There are three bags B 1 , B 2 and B 3 . The bag B 1 contains 5 red and 5 green balls, B 2 contains 3 red and 5 green balls, and B 3 contains 5 red and 3 green balls, Bags B 1 , B 2 and B 3 have probabilities 3 10 , 3 10 and 4 10 respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?

Probability that the chosen ball is green, given that the selected bag is B 3 , equals 3 8

Probability that the selected bag is B 3 and the chosen ball is green equals 3 10

Probability that the selected bag is B 3 , given that the chosen balls is green, equals 5 13

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Three urns are given, each containing red and black balls as indicated below:

Urn $I$ $6$ red and $4$ black balls

Urn $II$ $2$ red and $6$ black balls

Urn $III$ $1$ red and $8$ black balls

An urn is chosen at random and a ball is drawn from that urn. The ball drawn is red. The probability that the ball is drawn either from urn $II$ or from urn $III$ is given by $\frac{x}{y}$ . Find $x + y$ .

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Important Questions on Probability

EASY

Mathematics>Algebra>Probability>Bayes' Theorem

Bag

I

contains

3

red and

4

black balls. Bag II contains

5

red and

6

black balls. If one ball is drawn at random from one of the bags, and it is found to be red, then the probability that it was drawn from Bag

II

, is

HARD

Mathematics>Algebra>Probability>Bayes' Theorem

Box

1

contains

30

cards numbered

1

30

and Box

2

contains

20

cards numbered

31

50

. A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box

1

HARD

Mathematics>Algebra>Probability>Bayes' Theorem

In a test an examiner either guesses or copies or knows the answer to a multiple choice question with

4

choices. The probability that he/she makes a guess is

\frac{1}{3}

. The probability that he/she copies the answer is

\frac{1}{6}

. If the probability that the answer is correct, given that he/she copied it is

\frac{1}{8}

, then the probability that he/she knows the answer to a question given that he/she correctly answered it, is

HARD

Mathematics>Algebra>Probability>Bayes' Theorem

The probabilities of having a defective toy in three cartons

A, B, C

are

\frac{1}{3}, \frac{1}{4}, \frac{2}{5}

respectively. If a carton is selected at random and a toy drawn randomly from it is found to be defective, then the probability that it is drawn from carton

B

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Mathematics>Algebra>Probability>Bayes' Theorem

A bag contains

4

red and

6

black balls. A ball is drawn at random from the bag, its color is observed and this ball along with two additional balls of the same color are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is:

HARD

Mathematics>Algebra>Probability>Bayes' Theorem

Events

E_{1}

and

E_{2}

form a partition of the sample space

S

A

is any event such that

P (E_{1}) = P (E_{2}) = \frac{1}{2}, P (E_{2} / A) = \frac{1}{2}

and

P (A / E_{2}) = \frac{2}{3}

, then

P (E_{1} / A)

MEDIUM

Mathematics>Algebra>Probability>Bayes' Theorem

Out of 11 consecutive natural number if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference is :

HARD

Mathematics>Algebra>Probability>Bayes' Theorem

A certain item is manufactured by

3

factories

F_{1}, F_{2}

and

F_{3}

with

30 %

of items made in

F_{1}, 20 %

F_{2}

and

50 %

F_{3}

. It is found that

2 %

of the items produced by

F_{1}, 3 %

of the items produced by

F_{2}

and

4 %

of the items produced by

F_{3}

are defective. Suppose that an item selected at random from the stock is found defective. What is the probability that the item came from

F_{1}

HARD

Mathematics>Algebra>Probability>Bayes' Theorem

Let

n_{1} & n_{2}

be the number of red and black balls, respectively, in box

I

. Let

n_{3} & n_{4}

be the number of red and black balls, respectively, in box

II

. One of the two boxes, box

I

and box

II

, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box

II

\frac{1}{3}

, then the correct option(s) with the possible values of

n_{1}, n_{2}, n_{3} & n_{4}

is(are)

HARD

Mathematics>Algebra>Probability>Bayes' Theorem

There are three bags

B_{1}, B_{2}

and

B_{3} .

The bag

B_{1}

contains

5

red and

5

green balls,

B_{2}

contains

3

red and

5

green balls, and

B_{3}

contains

5

red and

3

green balls, Bags

B_{1}, B_{2}

and

B_{3}

have probabilities

\frac{3}{10}, \frac{3}{10}

and

\frac{4}{10}

respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?

HARD

Mathematics>Algebra>Probability>Bayes' Theorem

An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered

1, 2, 3, \dots, 9

is randomly picked and the number on the card is noted. The probability that the noted number is either

7

8

MEDIUM

Mathematics>Algebra>Probability>Bayes' Theorem

A box contains

b

blue balls and

r

red balls. A ball is drawn randomly from the box and is returned to the box with another ball of the same colour. The probability that the second ball drawn from the box is blue, is

HARD

Mathematics>Algebra>Probability>Bayes' Theorem

The probability of men getting a certain disease is

\frac{1}{2}

and that of women getting the same disease is

\frac{1}{5} .

The blood test that identifies the disease gives the correct result with probability

\frac{4}{5}

. Suppose a person is chosen at random from a group of

30

males and

20

females, and the blood test of that person is found to be positive. What is the probability that the chosen person is a man?

MEDIUM

Mathematics>Algebra>Probability>Bayes' Theorem

A company produces

10, 000

items per day. On a particular day

2500

items were produced on machine

A

3500

on machine

B

and

4000

on machine

C

. The probability that an item produced by the machines

A, B, C

to be defective is respectively

2 %, 3 %

and

5 %

. If one item is selected at random from the output and is found to be defective, then the probability that it was produced by machine

C

, is

HARD

Mathematics>Algebra>Probability>Bayes' Theorem

Ravi and Rashmi are each holding

2

red cards and

2

black cards (all four red and all four black cards are identical). Ravi picks a card at random from Rashmi and then Rashmi picks a card to random from Ravi. This process is repeated a second time. Let

p

be the probability that both have all

4

cards of the same colour. Then,

p

satisfies

HARD

Mathematics>Algebra>Probability>Bayes' Theorem

In medical diagnostics for a disease, sensitivity (denoted a) of a test refers to the probability that a test result is positive for a person with the disease whereas specificity (denoted b) refers to the probability that a person without the disease test negative. A diagnostic test for influenza has the values of

a = 0.9

and

b = 0.9 .

Assume that the prevalence of influenza in a population in

50 % .

If a randomly chosen person tests negative, what is the probability that the person actually has influenza?

MEDIUM

Mathematics>Algebra>Probability>Bayes' Theorem

In medical diagnostics for a disease, sensitivity (denoted by

a

) of a test refers to the probability that a test result is positive for a person with the disease, whereas specificity (denoted by

b

) refers to the probability that a person without the disease tests negative. A diagnostic test for

COVID - 19

has the values of

a = 0.99

and

b = 0.99

. If the prevalence of

COVID - 19

in a population is estimated to be

10 %

, what is the probability that a randomly chosen person tests positive for

COVID - 19

MEDIUM

Mathematics>Algebra>Probability>Bayes' Theorem

A man speaks truth 2 out of 3 times. He picks one of the natural numbers in the set $S = {1, 2, 3, 4, 5, 6, 7}$ and reports that it is even. The probability that it is actually even is

HARD

Mathematics>Algebra>Probability>Bayes' Theorem

Two persons

A

and

B

throw a (fair) die

(

six-faced cube with faces numbered from

1

6

)

alternately, starting with

A

. The first person to get an outcome different from the previous one by the opponent wins.The probability that

B

wins is,

HARD

Mathematics>Algebra>Probability>Bayes' Theorem

A computer producing factory has only two plants $T_{1}$ and $T_{2}$ . Plant $T_{1}$ produces $20 %$ and plant $T_{2}$ produces $80 %$ of the total computers produced. $7 %$ of computers produced in the factory turn out to be defective. The probability that a computer turns out to be defective which is produced in plant $T_{1}$ is ten times of the computers produced in the plant $T_{2}$ . A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant $T_{2}$ is

Urn $I$	$6$ red and $4$ black balls
Urn $II$	$2$ red and $6$ black balls
Urn $III$	$1$ red and $8$ black balls

Important Questions on Probability

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