Question 1

What is the probability that an ordinary year has 53 Sundays?

Accepted Answer

An ordinary year has 365 days i,e. 52 weeks and 1 day. So the probability that this day is a Sunday is 1/7.

Question 2

One card is drawn at random from a pack of 52 cards. What is the probability that the card drawn is a face card?

Accepted Answer

Clearly n(S) = 52 and there are 16 face cards. ∴ P(E) = 16/52 = 4/13

Question 3

One card is drawn at random from a pack of 52 cards. What is the probability that the card drawn is either a red card or a king?

Accepted Answer

Clearly n(S) = 52. There are 26 red cards (including 2 kings ) and there are 2 more kings. Let (E) be the event of getting either a red card or a king . Then, n(E) = 28 ∴ P(E) = n(E) / n(S) = 28/52 = 7/13

Question 4

What if the probability of getting a king or a queen in a single drawn from a pack of 52 cards?

Accepted Answer

Clearly, n (S) = 52, there are 4 kings and 4 queens ∴ P(E) = n (E) / n (S) = 8/52 = 2/13

Question 5

A card is drawn from a pack of 52 cards. A card is drawn at random. What is the probability that it is neither a heart nor a king?

Accepted Answer

There are 13 hearts and 3 more kings ∴ p (heart or a king ) = (13 + 3)/52 = 4/13 ∴ P (neither a heart nor a king) = 1 - 4/13 = 9/13

Question 6

Seven white balls and three black balls are randomly placed in a row, The probability that no two black balls are placed adjacently equals?

Accepted Answer

First we place seven white balls at places market W. W -- W --W --W --W -- W -- W If we place three black balls at -- places. Then no two black balls will be placed adjacently. Total No, of -- places = 8 ∴ No. of favourable ways for black balls = 8C3 = 8 x 7 x 6 / 1 x 2 x 3 = 56 Total no. of equally likely cases = 10! / 7!3! = (10 x 9 x 8) / (1 x 2 x 3) = 15 x 8 = 120 ∴ Reqd. Probability = 56 / 126 = 7/15

Question 7

The chance of throwing a total of 3 or 5 or 11 with two dics is?

Accepted Answer

∵ Probability for 3 = (1, 2), (2, 1) = 2/36 ⇒ Probability for 5 = (1, 4), (2, 3), (3, 2), (4, 1)= 4/ 36 ⇒ Probability for 11 (5, 6), (6, 5) = 2/36 ∴ Reqd . Probability = 2/36 + 4/36 + 2/36 = 8/36 = 2/9

Question 8

The probability that a man lives after 10 years is 1/4 and that his wife is alive after 10 years is 1/3. The probability that neither of them is alive after 10 years is?

Accepted Answer

P(M) = 1/4; P(W)= 1/3 ⇒ P (M) = 1 - 1/4 = 3/4 ⇒ P(W)= 1 - 1/3 = 2/3 Reqd. probability = P(M) P(W) = 3/4 x 2/3 = 1/2

Question 9

The probability that a person will hit a target in shooting practice is 0.3 . If he shoots 10 times. Then the probability of his shooting the target is?

Accepted Answer

∵ Probability in each trial (shooting) = 0.3 ∴ Reqd. probability = (0.3)10

Question 10

The probability that a marksman will hit a target is given as 1/5. Then his probability of at least one hit in 10 shots, is?

Accepted Answer

∵ p = 1/5 ∴ q = 1 - 1/5 = 4/5 The probability that none will hit in 10 shots = (4/5)10 ∴ Reqd. probability = 1 - (4/5)10

Question 11

The number of tosses that have to be made in order that there is 99% probability of getting at least one head is?

Accepted Answer

Corresponding to n tosses, the Probability of getting no head = (1/2)n and, Therefore, the probability of getting at least one head = 1 - (1/2)n Now, 1 - (1/2)n ≥ 99/100 ⇒ (1/2)n ≤ 1/100 ⇒ n ≥ 7

Question 12

If an integer is selected at random from 1 to 100. The probability that it relatively prime to 100 is?

Accepted Answer

From the set of odd number < 100, if we exclude multiples of 5, we get the set of numbers < 100 and relatively prime to 100. The number of such numbers = 50 - 10 = 40 ∴ Reqd. Probability = 40 / 100 = 2/5

Question 13

n books are to be arranged on a self. These include m volumes of a science book (m < n). The probability that in any arrangement, the volumes of science books are in ascending order is?

Accepted Answer

Corresponding to each arrangement of (n - m) other books, there is a unique arrangement of the m volumes of the science book in ascending order and m! arrangement of the m volumes in random order &Reqd. Prob = p = 1/m!

Question 14

The probability that a man aged x years will die in a years is p. The probability that out of n men M1, M2, M3...... Mn each aged n years Mk will die and be the first to die is?

Accepted Answer

∵ Probability that no man out of n men aged x years will die in a year = (1 - p)n ∴ Probability that out of n men at least one will die in a year = 1 - (1 - p)n . When at least one, man dies, any one out of the n men may be the first to die

Question 15

If the probability of machine failing during a day is 0.95. The probability of its working for four consecutive days without failing is?

Accepted Answer

∴ Probability of machine failing during a day p = 0.95 ∴ q = Probability of its working during a day = 1 - p = 1 - 0.95 = 0.05 Required probability = q4 = (0.05)4 = 0.00000625

Question 16

Seven white balls and three black ball are randomly placed in a row. The probability that no two black balls are placed adjacently equals?

Accepted Answer

n(S) = 10C7 = 10C3 n(E) = 8C3 ∴ P(E) = n(E)/n(S) = 7/15

Question 17

The probability of occurrence of a multiple of 2 on a dice and multiple of 3 on the other dice . If both are thrown together is?

Accepted Answer

Probability of multiple of 2 = 3/6 = 1/2 Probability of multiple of 3 = 2/6 = 1/3 Since there are two dice. ∴ The required probability = 2 x 1/2 x 1/3 = 1/3

Question 18

Two number are selected randomly from the set S = {1, 2, 3, 4, 5, 6} without replacement one by one. The probability that minimum of the number is less than 4 is?

Accepted Answer

Total no . of cases = 6P2 = 6 x 5 = 30 Non-favourable cases are (4, 5) , (5, 4), (4, 6), (6, 4), (5, 6), (6, 5) ∴ Probability that event will not happen = 6 /(6 x 5) = 1/5 ∴ Reqd. probability = 1 - 1/5 = 4/5

Question 19

A bag contains 5 blue and 4 black balls. Three balls are drawn at random. What is the probability that 2 are blue and 1 is black?

Accepted Answer

Let S be the sample space and E be the event of drawing 3 balls out of which 2 are blue and 1 is back . Then, n (S)= Number of ways of drawing 3 balls out of 9=9c3 = (9 x 8 x 7) / (3 x 2 x 1) = 84 and n(E) = Number of ways of drawing 2 balls out of 5 and 1 ball out of 4 . = 5C2 4C11 = (5 x 4) / (2 x 1) + 4 = 14 ∴ P(E) = n(E)/n(S) = 14/84 = 1/6

Question 20

A six - faced dice is so biased that is twice as likely to show an even number as an odd number when throw. It is thrown twice. The probability that the sum of two number thrown is even is?

Accepted Answer

∵ Probability for odd = p ∴ Probability for even = 2p ∵ p + 2p = 1 ⇒ 3p = 1 ⇒ p = 1/3 ∴ Probability for odd = 1/3, Probability for even = 2/3. Sum of two nos. is even means either both are odd or both are even ∴ Reqd. probability = 1/3 x 1/3 + 2/3 x 2/3 = 1/9 + 4/9 = 5/9 [ ∵ die is thrown twice ]

Probability Questions