When MS and MBA applicants ask us – ‘*What are my chances of getting into Harvard?*‘ or ‘*What’s my probability of getting scholarships from Oxford?*‘ we get tongue-tied. There are so many variables at play, it’s difficult to give an accurate answer.

But when you get probability questions in your GRE and GMAT exam syllabus, you don’t have to get flummoxed. Understanding the basic rules and formulas of probability will help you score high in the entrance exams.

As the Oxford dictionary states it, Probability means ‘The extent to which something is probable; the likelihood of something happening or being the case’.

In mathematics too, probability indicates the same – the likelihood of the occurrence of an event.

Examples of events can be :

- Tossing a coin with the head up
- Drawing a red pen from a pack of different coloured pens
- Drawing a card from a deck of 52 cards etc.

Either an event will occur for sure, or not occur at all. Or there are possibilities to different degrees the event may occur.

An event that occurs for sure is called a Certain event and its probability is 1.

An event that doesn’t occur at all is called an impossible event and its probability is 0.

This means that all other possibilities of an event occurrence lie between 0 and 1.

This is depicted as follows:

**0 <= P(A) <= 1**

where A is an event and P(A) is the probability of the occurrence of the event.

This also means that a probability value can never be negative.

Every event will have a set of possible outcomes. It is called the ‘sample space’.

Consider the example of tossing a coin.

When a coin is tossed, the possible outcomes are Head and Tail. So, the sample space is represented as {H, T}.

Similarly when two coins are tossed, the sample space is {(H,H), (H,T), (T,H), (T,T)}.

The probability of head each time you toss the coin is 1/2. So is the probability of tail.

As you might know from the list of GMAT maths formulas, the Probability of the occurrence of an event A is defined as:

Another example is the rolling of dice. When a single die is rolled, the sample space is {1,2,3,4,5,6}.

What is the probability of rolling a 5 when a die is rolled?

No. of ways it can occur = 1

Total no. of possible outcomes = 6

So the probability of rolling a particular number when a die is rolled = 1/6.

Compound probability is when the problem statement asks for the likelihood of the occurrence of more than one outcome.

- P(A or B) = P(A) + P(B) – P(A and B)

where A and B are any two events.

P(A or B) is the probability of the occurrence of atleast one of the events.

P(A and B) is the probability of the occurrence of both A and B at the same time.

Mutually exclusive events are those where the occurrence of one indicates the non-occurrence of the other

OR

When two events cannot occur at the same time, they are considered mutually exclusive.

__Note:__ For a mutually exclusive event, P(A and B) = 0.

**Example 1: **What is the probability of getting a 2 __or__ a 5 when a die is rolled?

__Solution:__

Taking the individual probabilities of each number, getting a 2 is 1/6 and so is getting a 5.

Applying the formula of compound probability,

Probability of getting a 2 **or **a 5,

P(2 or 5) = P(2) + P(5) – P(2 and 5)

==> 1/6 + 1/6 – 0

==> 2/6 = 1/3.

**Example 2: **Consider the example of finding the probability of selecting a black card or a 6 from a deck of 52 cards.

__Solution:__

We need to find out P(B or 6)

Probability of selecting a black card = 26/52

Probability of selecting a 6 = 4/52

Probability of selecting both a black card and a 6 = 2/52

P(B or 6) = P(B) + P(6) – P(B and 6)

= 26/52 + 4/52 – 2/52

= 28/52

= 7/13.

When multiple events occur, if the outcome of one event __DOES NOT__ affect the outcome of the other events, they are called independent events.

Say, a die is rolled twice. The outcome of the first roll doesn’t affect the second outcome. These two are independent events.

**Example 1: **Say, a coin is tossed twice. What is the probability of getting two consecutive tails ?

Probability of getting a tail in one toss = 1/2

The coin is tossed twice. So 1/2 * 1/2 = 1/4 is the answer.

Here’s the verification of the above answer with the help of sample space.

When a coin is tossed twice, the sample space is {(H,H), (H,T), (T,H), (T,T)}.

Our desired event is (T,T) whose occurrence is only once out of four possible outcomes and hence, our answer is 1/4.

**Example 2: **Consider another example where a pack contains 4 blue, 2 red and 3 black pens. If a pen is drawn at random from the pack, replaced and the process repeated 2 more times, What is the probability of drawing 2 blue pens and 1 black pen?

**Solution**

Here, total number of pens = 9

Probability of drawing 1 blue pen = 4/9

Probability of drawing another blue pen = 4/9

Probability of drawing 1 black pen = 3/9

Probability of drawing 2 blue pens and 1 black pen = 4/9 * 4/9 * 3/9 = 48/729 = 16/243

When two events occur, if the outcome of one event affects the outcome of the other, they are called dependent events.

Consider the aforementioned example of drawing a pen from a pack, with a slight difference.

**Example 1:** A pack contains 4 blue, 2 red and 3 black pens. If 2 pens are drawn at random from the pack, __NOT__ replaced and then another pen is drawn. What is the probability of drawing 2 blue pens and 1 black pen?

**Solution:**

Probability of drawing 1 blue pen = 4/9

Probability of drawing another blue pen = 3/8

Probability of drawing 1 black pen = 3/7

Probability of drawing 2 blue pens and 1 black pen = 4/9 * 3/8 * 3/7 = 1/14

Let’s consider another example:

**Example 2:** What is the probability of drawing a king and a queen consecutively from a deck of 52 cards, __without__ replacement.

Probability of drawing a king = 4/52 = 1/13

After drawing one card, the number of cards are 51.

Probability of drawing a queen = 4/51.

Now, the probability of drawing a king and queen consecutively is 1/13 * 4/51 = 4/663

Conditional probability is calculating the probability of an event given that another event has already occured .

The formula for conditional probability P(A|B), read as P(A given B) is

Consider the following example:

**Example:** In a class, 40% of the students study math and science. 60% of the students study math. What is the probability of a student studying science given he/she is already studying math?

**Solution**

P(M and S) = 0.40

P(M) = 0.60

P(S|M) = P(M and S)/P(S) = 0.40/0.60 = 2/3 = 0.67

A complement of an event A can be stated as that which does NOT contain the occurrence of A.

A complement of an event is denoted as P(A^{c}) or P(A’).

**or it can be stated, P(A)+P(A ^{c}) = 1**

For example,

if A is the event of getting a head in coin toss, A^{c} is not getting a head i.e., getting a tail.

if A is the event of getting an even number in a die roll, A^{c }is the event of NOT getting an even number i.e., getting an odd number.

if A is the event of randomly choosing a number in the range of -3 to 3, A^{c} is the event of choosing every number that is NOT negative i.e., 0,1,2 & 3 (0 is neither positive or negative).

Consider the following example:

**Example:** A single coin is tossed 5 times. What is the probability of getting at least one head?

__Solution:__

Consider solving this using complement.

Probability of getting no head = P(all tails) = 1/32

P(at least one head) = 1 – P(all tails) = 1 – 1/32 = 31/32.

What is the probability of the occurrence of a number that is odd or less than 5 when a fair die is rolled.

**Solution**

Let the event of the occurrence of a number that is odd be ‘A’ and the event of the occurrence of a number that is less than 5 be ‘B’. We need to find P(A or B).

P(A) = 3/6 (odd numbers = 1,3 and 5)

P(B) = 4/6 (numbers less than 5 = 1,2,3 and 4)

P(A and B) = 2/6 (numbers that are both odd and less than 5 = 1 and 3)

Now, P(A or B) = P(A) + P(B) – P(A or B)

= 3/6 + 4/6 – 2/6

P(A or B) = 5/6.

A box contains 4 chocobars and 4 ice creams. Tom eats 3 of them one after another. What is the probability of sequentially choosing 2 chocobars and 1 icecream?

**Solution**

Probability of choosing 1 chocobar = 4/8 = 1/2

After taking out 1 chocobar, the total number is 7.

Probability of choosing 2nd chocobar = 3/7

Probability of choosing 1 icecream out of a total of 6 = 4/6 = 2/3

So the final probability of choosing 2 chocobars and 1 icecream = 1/2 * 3/7 * 2/3 = 1/7

When two dice are rolled, find the probability of getting a greater number on the first die than the one on the second, given that the sum should equal 8.

**Solution**

Let the event of getting a greater number on the first die be G.

There are 5 ways to get a sum of 8 when two dice are rolled = {(2,6),(3,5),(4,4), (5,3),(6,2)}.

And there are two ways where the number on the first die is greater than the one on the second given that the sum should equal 8, G = {(5,3), (6,2)}.

Therefore, P(Sum equals 8) = 5/36 and P(G) = 2/36.

Now, P(G|sum equals 8) = P(G and sum equals 8)/P(sum equals 8)

= (2/36)/(5/36)

= 2/5

Probability Problem 1

A bag contains blue and red balls. Two balls are drawn randomly without replacement. The probability of selecting a blue and then a red ball is 0.2. The probability of selecting a blue ball in the first draw is 0.5. What is the probability of drawing a red ball, given that the first ball drawn was blue?

a) 0.4

b) 0.2

c) 0.1

d) 0.5

Answer 1

A.

Problem 2

A die is rolled thrice. What is the probability that the sum of the rolls is atleast 5.

a) 1/216

b) 1/6

c) 3/216

d) 212/216

Answer 2

D.

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## 64 Comments

a number of people gave a hat check girl one hat. suppose all the tickets got misplaced, so all the hat were given back randomly.

a) if its 2 people determine the probability at least one person got their hat returned.

b) if its 3 people determine the probability at least one person got their hat returned.

c) if its 4 people determine the probability at least one person got their hat returned.

d) if its 5 people determine the probability at least one person got their hat returned.

Hi I’m Algia and I need help in solving this problem, can you help me please.

a) 1-(0.5)=0.5

b) 1-(0.667*0.5)=0.667

c) 1-(0.75*0.667*0.5)=0.75

d) 1-(0.8*0.75*0.667*0.5)=0.8

Really nice sequence.

P(h’)=1-P(h) etc. At least one hat is correctly returned is compliment that no hat is returned correctly.

(n-1)/n

The math here is totally wrong

Feel free to enlighten!

Hi

Last question must be 212/216 right ?

Tell me the way u did that sol. Plzz…

yes its must be 212/216

I think it should be 212/216

Bcz we have 4 number of event to getting a number of sum less than 5

{(1,1,1),(1,1,2),(1,2,1),(2,1,1)}

It means p(e) = 4/216

Nd getting a number of sum at least 5

Is

1-4/216=212/216

you are telling a wrong way so stop telling me your bloody wrong answer it’s a correct answer it’s should not be 212/216..

A man answers 10 maths problems, one after the other. He answers the first problem correctly and the second problem incorrectly, for each of the remaining 8 problems the probability that he answers the problem correctly equals to the ratio of the number of problems that he has already answered correctly to the total number of problems that he has already answered. What is the probability that he answers exactly 5 out of 10 problems correctly??

A woman bought 5basket of tomatos each costing 1250naira,in her discovery she observe that 90% of the tomatos where damage resulting to a loss of 510naira.(a)what is the probability of obtaining an average of 50 if the cost per bag is 50 above the cost?(b)what will be the actual price for selling the tomato at cost plus(+) 25%?

Simple way:

Q: A die is rolled thrice. What is the probability that the sum of the roll is at least 5?

A: At least 5 is equal or greater 5 ===> which is P(x> or =5)= 1-P(x<5)

Then: P(x<5)= 4/216. . .taken thrice rolls =6*6*6=216

Therefore: 1-P(x 53/54

the personal director of a company wishes to select applicant for advanced training without regard to sex. let “W” denotes women and “M” denotes men and the pattern of arrival be M WWW MMM WW M WWW MMMM W M W MM WWW MM W MMMM WW M WW MMMM WW M WWWW MM WW M W WW. will you conclude that the applicants have arrived in a random fashion?

pavement.before any 250 m length of a pavement is accepted by the state highway department,the thickness of a30 m s mointored by an altrasonic to verify compliance to specification .each section is rejected if a measurment thickness less than 10cm;otherwise the all section is accepted .from past experment ,the stat highway engineer know the 85%of all section constructed by the contructor comply with specification . however the relability of altrusonic thickness testing is only 75 ,so that there is a 25 percent chane of errorneous concolusion based on the determenation of thickness with ultrasonic . what is the probablity that a poorly constructed section is accepted on the base of ultrasonic test?

The chance or probability of getting accepted is 0.85; the chance of getting accepted even when bad is 0.25. So therefore the chance of being bad and getting selected can be solved using the conditional probability theorem given by:

P(A/B)= P(AnB)/P(B).

Going by this the answer is: 0.25 x 0.85= 0.2125

solution

the possible out come of rolling die is =6 here in this case since it is rolled 3 our sample space is 6×6×6=216

we have asked to solve the probability of sum which will be atleast 5 this means 5 and more is possible. so that we have to search the possibilities of less than five to easy our work this will be like[111][112][121] = 3 out comes onlywso p(s`)=3/216 when p(s`) is probability of sum less than five or probability of sum greater than equal to five.

since the sum of p(s) and p(s`)=1

p(s)=1-p(s~)

1-3/216=213/216

what about 2,1,1?

Two cards are drawn at random from an ordinary deck of 52 card. Find the probability P that

(a) Both are spade

(b) One is a spade and one is heart

Ans:

(a) Probability of getting spade 1st time is 13/52 and Probability of getting spade 2nd time is is 12/51

Total Probability is 13*12/(52*51) = 156/2652

(b) Probability of getting spade is 13/52 and Probability of getting Heart is 12/51

Total probability is 13*13/(52*51) = 169/2652

Question ‘b’ says that one is a spade and one is a heart. Therefore the possibilities are ‘heart-spade’ and ‘spade-heart’.

The answer should be:

((13*13/52*51) + (13*13/52*51)) = 169/2652 + 169/2652 =338/2652 = 13/102

copying the solution offerred by @ diriba

solution

the possible out come of rolling die is =6 here in this case since it is rolled 3 our sample space is 6×6×6=216

we have asked to solve the probability of sum which will be atleast 5 this means 5 and more is possible. so that we have to search the possibilities of less than five to easy our work this will be like[111][112][121] = 3 out comes onlywso p(s`)=3/216 when p(s`) is probability of sum less than five or probability of sum greater than equal to five.

since the sum of p(s) and p(s`)=1

p(s)=1-p(s~)

1-3/216=213/216

The above solution is good but a little faulty because it considered only the possibility of obtaining a ‘1’ on the first die, it omitted the possibility of getting a ‘2’ on the first die i.e (using the same notation) [211], this is the fourth possible outcome.

Hence P(s)= 1- P(s’)

= 1-4/216

=212/216

=53/54

A bag contains blue and red balls. Two balls are drawn randomly without replacement. The

probability of selecting a blue and then a red ball is 0.2. The probability of selecting a blue ball in the first

draw is 0.5. What is the probability of drawing a red ball, given that the first ball drawn was blue? Solution please

Lets assume probability of picking a red ball is X.

The probability of selecting a blue ball and then a red ball,

P(B)*P(R)=.2

.5*X=.2

x=.5/.2

x=.4

P(R|B)=P(R and B)/P(B)

=0.2/0.5=0.4

The personal director of a company wishes to select applicant for advanced

training without regard to sex. Let ‘W’ denote Women and ‘M’ Denotes men and

the pattern of arrival be M WWW MMM WW M WWW MMMM W M W MM WWW

MM W MMMM WW M WW MMMM WW M WWWW MM WW M W M WW. Will

you conclude that the applicants have arrived in a random fashion?

The probability of snow tomorrow is 0.6. And the probability that it will bi colder is 0.7. The probability that it will not snow and not bi colder is 0.1 .What is probability that it will not snow if it is colder tomorrow?

please solve it … and tell me answer.. thanks …

A= event for it will snow tomorrow

B= event for it will be cold

a= event for it will not snow

b= event for it will not be cold

P(A) = 0.6; P(B)= 0.7; P(a n b)=0.1

P(A)+P(B)-P(AnB)+P(a n b)=1

Inserting values, you’ll have

P(AnB)=0.4.

P(Bna)= P(B)-P(BnA)

This will give you P(Bna)= 0.3.

By conditional theorem,

P(a/B)=P(anB)/P(B)

This will give you 3/7 as the answer.

pl first draw a tree diagram:

P(S)=0.6, P(N.S.)=0.4

P(Cold)=0.7, P(Not Cold)=0.3

P(No S + Not Cold)= 0.4*0.3=0.12=0.1

P(No S + Cold)=0.4*0.7=0.28

in a class 10 boys and 5 girls .three students are selected random one after the other.find the probability that

1)first two are boys and third is girl

2)first and third is of same gender and third is of opposite gender

please help me in solving this

A) 10/15*9/14*5/13

B) 1st case:

1st & 2nd are boys & 3rd is girl

10/15*9/14*5/13

2nd case:

1st & 2nd are girls & 3rd is boy

5/15*4/14*10/13

n(b)=10

n(g)=5

n(t)=15

(a) p(first two are boys and third is a girl)

p(B,B,G)

10/15*9/14*5/13=

5/1*3/7*1/13=15/91

(b) p(first and third is of the same gender and second is opposite)

P(B,G,B) or p(G,B,G)

(10/15*5/14*9/13)+(5/15*10/14*4/13)=

(450/2730)+(200/2730)=

450+200/2730=

650/2730=

5/21

A boy goes to his school either by bus or on foot. If one day he goes to the school by bus, then the probability that he goes by bus the next day is 7/10. If one day he walks to the school, then the probability that he goes by bus the next day is 2/5.

i) Given that he walks to the school on a particular Tuesday, find the probability that he will go to the school by bus on Thursday of that week.

ii) Given that the boy walks to the school on both Tuesday and Thursday of that week, find the probability that he will also walk to the school on Wednesday.

There are three boxes, one of which contains a prize. A contestant is given two chances, such that if he chooses the wrong box in the first round, that box is removed from the selection and he then chooses between the two remaining boxes.

1. What is the probability that the contestant wins?

2. Does the contestant’s probability of winning increases on the second round?

Hi Lei,

It’s a Monty Hall problem. You can google it.

As for your question,

As the first box chosen if found empty is removed and you HAVE/Switch to pick from other two, the P(W) = 2/3.

Above answer can be explained as Prob. of winning on first box + Prob. of choosing wrong * Prob. of Choosing right between the two => 1/3+2/3*1/2 => 2/3

The answer to the second: Yes probability increases as its a 50% chance to win as 1 wrong box is eliminated.

1) 10C2*5C1/15C3?

2) (10C1*5C1*9C1/15C3) + (5C1*10C1*4C1/15C3)?

Plz solve it

XYZ company wants to start a food outlet in pakistan. There is a 40% and 60% chance of stating in hyderabad and karachi respectively. If he start the outlet in hyderabad there is 30% chance that it will be in saddar and 70% chance that it will be in defence area. If they start the outlet in karachi there is 50% chance that it will be in defence, 30% in clifton and 20% in pechs. Determine probability of starting the outlet in: (a) saddar (b) defence area of any city (c) clifton given that the outlet is started in karachi

a) P(H,S) = 40% x 30% =0.4 x 0.3 = 0.12 = 12%

b) P(H, D) + P(K, D) = 40% x 70% + 60% x 50% = 0.4 x 0.7 + 0.6 x 0.5 = 0.28 + 0.3 = 0.58 = 58%

c) P(C|K) = 30%

please solve these questions. 1. The probability that a randomly chosen sales prospect will make a purchase is 0.20. if a sales man calls an 6 prospects, what is the probability that he will make ……….. a) exactly 4 sales b) 4 or more sales c) no sales

a) 15*(0.2)^4*(0.8)^2 = 0.01536

b) 0.01536 + 6*(0.2)^5*(0.8)^1 + 1*(0.2)^6*(0.8)^0 = 0.01696

c) (0.8)^6 = 0.262144

please solve this problem : Suppose 100 new born in a maternity clinic , 55 were females and 45 males . What is the probability of the next three deliveries are females ?

In maternity clinic the probability of new born was females is 55%=0.55

So,the probabilitt of the next three deliveries are females is 0.55×0.55×0.55=0.166 or 16.6%

Pls. Answer. Thanks. Five hundred raffle tickets are sold at P25 each for 3 pieces of P4,000, P250 and P1,000. After each price drawing, the winner is then returned to the collection of tickets. What is the expected value if the person purchases four (4) tickets?

a major urban hospital has gathered data on the number of heart attack victims seen.the given table indicates the probabilities of different numbers of heart attack victims being treated in the emergency room on a typical day number of victims treated (n)fever than 5 ,6,7,more than 7 p(n) 0.08,0.16,0.30,0.26,0.20

I have a question it goes a manufacturing firm produces units of products in 4 plants A,B,C and D from the first records of proportion of defectives produced at each brands following conditional probabilities are set A=0.5 B=1.0,C=0.15, D=0.02 the first plant produces 30% of the units of the output the 2nd plant produces 25% the 3rd produces 40% and the 4th 5% .a unit of the products meant at one of this plants is tested and its found to be defective. What’s the probability that the units was produced in plant C.

I am kindly asking for help with the below question

There are three routes from a person,s home to her place of work.there are four parking lots where she works, three entrances into her building, two elevators to her floor and one route from each elevator to her office door.

1. How many ways can she go from her home to her office?

2. If she makes her various choices at random,what is the probability that she will take mornungside drive,park in lot A,use the south entrance and take elevator 1.

3. As she starts her car one morning, she recalls parking lot A and B are closed for repair.what is the probability that she will take industrial avenue, park in lot D, use the north entrance and take elevator 2.

Please solve the following three questions for me.

1. A committee of 3 people was chosen at random from a group of 5 men and 6 women. Find the probability that the committee consisted of more men than women.

2. Two integers, x and y are selected at random from the integers 1 to 8. If the same integers may be selected twice, find the probability that:

a) |x-y|= 2

b) |x-y| is 5 or more.

c) x > y

3. A die is biased so that when tossed, the probability of a number r showing up, is given by P(r) = Kr where K is a constant and r = 1, 2, 3, 4, 5, 6 (the numbers on the faces of the die.

a) Find the value of K

b) If the die is tossed twice, calculate the probability that the total score is 11.

Please answer this question:

Three people get on an elevator that stops at three floors. Assuming that each has an equal probability of going to any one floor.

a) Find the probability that at none gets off at the first floor.

b) Find the probability that they all get off at different floors.

c)Find the probability that they all get off at the same floor.

hopefully you haven’t been waiting over a year for a response :) Here you go though…

A: (2/3)^ 3 = 8/27

B: 1 * 2/3 * 1/3 = 2/9

C: 1 * 1/3 * 1/3 = 1/9

3 balls are thrown one after the other with replacement frm a bag containing 5 red, 9 white and 4 blue identical balls.What is the probability that they are one red, one white and one blue

Hi pls solve this;

Two fair dice are rolled simultaneously.

(a) write the outcome table.

(b) find the probability of obtaining :

(i)equal number on the dice.

(ii) sum of at least 9.

(iii) sum of 10 or 4.

There are 18 balls in the bag .

The probability of choosing one red ball is 5/18.

We can choose the balls with replacement so that means that after choosing one red ball there are still 18 balls in the bag. the probability of choosing one whiteball is 9/18=1/2.

There are still 18 balls.

The probability of choosing one blue ball is 4/18=2/9.

The probability of choosing one red ball, one white ball and one blue ball is equal to 5/18*1/2*2/9

Three guys from Angie, Bessom and kugwe enter a raffle ( parifoot).The guy from Angie buys 3 tickets, the guy from Bessom buys 2 tickets and Kugwe buys 1. There are a total of 100 tickets in the raffle and 5 winning numbers are drawn . What is the probability of all three guys having at least a winning number ?

Please solve the question…

Dr. Stallter has been teaching basic statices for many years. She knows that 80% of the studentswill complete the assigned problem. She has also determined that among those who do their assignment, 90% will pass the cours. Among those studentswho do not do their homewor, 60% will pass. Mike Fishbaugh took statices last semester from Dr. Stallter and received a passing grade. What is the probability that he completed the assignment?

Can you solve me these 4 question???

1. A piece of medical equipment will function only when all the three components X, Y and Z are working. The probability X failing during one year is 0.25 and that of Y is 0.05 and that of Z failing is 0.15. what is the probability that the equipment will fail before the end of one year?

2. The probability that medical specialist will remain with a hospital is 0.6. The probability that an employee earns more than 40,000 per month is 0.5. The probability that an medical specialist earns more than 40,000 who remained with the hospital or more earns more than 40,000 per month given that he is medical specialist who stayed with the hospital?

3. A first aid box contains 8 packets of electrolyte in orange flavor and 5 packets of electrolyte in other flavor. Two successive drawings of 3 packets of electrolyte are made such that (i) electrolyte are replaced before second trial (ii) electrolyte are not replaced before second trial. Find the probability that the first drawing will give 3 packets of orange flavor and the second 3 other flavor.

4. In an examination there are objective and subjective questions. To each question, there are three possible answer and of which one is correct. An intelligence student knows 80% of the answer while a weak students knows only 10%.

(a) An intelligent students gets correct answer. What is the probability that he is guessing?

(b) If a weak student get the correct answer. what is the probability that he is guessing?

bits are sent over a communication channel in packets of 12. the probability of a bit being corrupted over this channel is 0.1 and such errors are independent. if 6 packets are sent over the channel what is the probability that at least one packet will contain 3 or more corrupted bits?

mr.x started his new business.The chance of Rs.40,000

profit per month is 20% and the chance of Rs,50,000

profits per month from the same business is 15%.

What is the probability that he will get a profit

of Rs.40,000 or Rs.50,000? how to solve this question

in a club with 8 men and 10 women members , find the probability that a 5 member committes chosen randomly will have a.all men b. 3 men and 2 women

An agency of the Rivers State Government is planning to invest in each of the LGA of the state. The probability that the product will be successful in a LGA is 0.65. The success in one LGA is independent of success or failure in the other LGA. What is the probability of success in at least two LGA.

Question 4 A new medical drug to cure HIV/AIDS has just been developed. The probability that it will cure HIV/AIDS is 0.9 if the patient has HIV/AIDS. The probability that it will cure HIV/AIDS falls to 0.4 if the patient does not have the disease. There is a probability of 0.3 that a person chosen at random from the community has HIV/AIDS. Find the following: i. Probability that a patient is cured of HIV/AIDS [12 Marks) Probability that a person is cured of HIV/AIDS actually had HIV/AIDS. [8 Marks) i

Question 1 [1, 1, 1, 3]

(a) Define a discrete random variable .

(b) Define the expected value of a discrete random variable .

(c) Define the variance of a discrete random variable .

(d) Let be a discrete random variable. Show that the [+]=2[].

Question 2 [3, 8]

(a) State the 5 properties required for a random variable to have a binomial distribution.

(b) Suppose that has a geometric distribution. Derive the mean and variance for the random variable Y

please help with the questions

please answer my question.

A ten digit number is formed using the digits from 0 to 9 and every digit being used only once.Find the probability that it is divisible by 4.

Need to get the correct answers please help me out!

2. In a group of a people, some are in favour of Biriyani and others are in favour of

Khichuri. Three (03) persons are selected at random from this group and asked whether

they are in favour of Biriyani or Khichuri. How many distinct outcomes are possible?

Draw a Venn diagram and a tree diagram for this experiment. List all the outcomes

included in each of the following events and mention whether they are simple or

compound events.

(a) All are in favour of the Khichuri.

(b) At most one person is in favour of Biriyani.

(c) Exactly one person is in favour of Khichuri.

P(A n B’) = 0,28 P(A’n B) = 0,12 P(A u B) = 0,68

Draw a Venn diagram illustrating this data.

Can someone solve this question?

Consider three English words, run, scythe, and breathe. The first can be used as noun and as verb, the second only as a noun, and the third only as a verb. For each word, we make a card with the word category of that word on the card. The card for run has Noun on one side, and Verb on the other. The cards for the other two words have the same word category label on either side. A card is drawn at random, and the word category label is read out load. If the label reads Verb, what is the probability that its associated word can also be used as a noun?

Thank you