ISC Class 12th Mathematics Chapter 13 - Probability Important Questions with Answers

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A box contains 30 fruits, out of which 10 are rotten. Two fruits are selected at random one by one without replacement from the box. Find the probability distribution of the number of unspoiled fruits. Also find the mean of the probability distribution.

Question 2.

A jewellery seller has precious gems in white and red colour which he has put in three boxes.

The distribution of these gems is shown in the table given below:

Box	Number of Gems
Box	White	Red
I	1	2
I	2	3
III	3	1

He wants to gift two gems to his mother. So, he asks her to select one box at random and pick out any two gems one after the other without replacement from the selected box. The mother selects one white and one red gem.

Calculate the probability that the gems drawn are from Box II.

Question 3.

A primary school teacher wants to teach the concept of 'larger number' to the students of Class II.

To teach this concept, he conducts an activity in his class. He asks the children to select two numbers from a set of numbers given as 2, 3, 4, 5 one after the other without replacement.

All the outcomes of this activity are tabulated in the form of ordered pairs given below:

	2	3	4	5
2	(2, 2)	(2, 3)	(2, 4)
3	(3, 2)	(3, 3)		(3, 5)
4	(4, 2)		(4, 4)	(4, 5)
5		(5, 3)	(5, 4)	(5, 5)

Complete the table given above.
Find the total number of ordered pairs having one larger number.
Let the random variable X denote the larger of two numbers in the ordered pair.
Now, complete the probability distribution table for X given below.
X 3 4 5
P(X = x)
Find the value of P(X < 5)
Calculate the expected value of the probability distribution.

Question 4.

Teena is practising for an upcoming Rifle Shooting tournament. The probability of her shooting the target in the 1^st, 2^nd, 3^rd and 4^th shots are 0.4, 0.3, 0.2 and 0.1 respectively. Find the probability of at least one shot of Teena hitting the target.

Question 5.

Five numbers x₁, x₂, x₃, x₄, x₅ are randomly selected from the numbers 1, 2, 3, ......., 18 and are arranged in the increasing order such that x₁ < x₂ < x₃ < x₄ < x₅. What is the probability that x₂ = 7 and x₄ = 11?

`26/51`

`3/104`

`1/68`

`1/34`

Question 6.

Given two events A and B such that (A/B) = 0.25 and P(A ? B) = 0.12. The value P(A ? B') is ______.

0.36

0.48

0.88

0.036

Question 7.

There are three machines and 2 of them are faulty. They are tested one by one in a random order till both the faulty machines are identified. What is the probability that only two tests are needed to identify the faulty machines?

Question 8.

Three friends go to a restaurant to have pizza. They decide who will pay for the pizza by tossing a coin. It is decided that each one of them will toss a coin and if one person gets a different result (heads or tails) than the other two, that person would pay. If all three get the same result (all heads or all tails), they will toss again until they get a different result.

What is the probability that all three friends will get the same result (all heads or all tails) in one round of tossing?
What is the probability that they will get a different result in one round of tossing?
What is the probability that they will need exactly four rounds of tossing to determine who would pay?

Question 9.

Students of under graduation submitted a case study on “Understanding the Probability of Left-Handedness in Children Based on Parental Handedness”. Following Recent studies suggest that roughly 12% of the world population is left-handed. Depending on the parents’ handedness, the chances of having a left-handed child are as follows:

Scenario A: Both parents are left-handed, with a 24% chance of the child being left-handed.

Scenario B: The fathers is right-handed and the mothers left-handed, with a 22% chance of child being left-handed.

Scenario C: The fathers left-handed and the mother is right-handed, with a 17% chance of child being left-handed.

Scenario D: Both parents are right-handed, with a 9% chance of having a left-handed child.

Assuming that scenarios A, B, C and D are equally likely and L denotes the event that the child is left-handed, answer the following questions.

What is the overall probability that a randomly selected child is left-handed?
Given that exactly one parent is left-handed, what is the probability that a randomly selected child is left-handed?
If a child is left-handed, what is the probability that both parents are left-handed?

Question 10.

Kiran plays a game of throwing a fair die 3 times but to quit as and when she gets a six. Kiran gets +1 point for a six and –1 for any other number.

If X denotes the random variable “points earned” then what are the possible values X can take?
Find the probability distribution of this random variable X.
Find the expected value of the points she gets.

Question 1.

A card from a pack of 52 playing cards is lost. From the remaining cards of the pack three cards are drawn at random (without replacement) and are found to be all spades. Find the probability of the lost card being a spade.

Question 2.

Evaluate P(A ? B), if 2P(A) = P(B) = `5/13` and P(A | B) = `2/5`

Question 3.

If A and B are events such as that P(A) = `1/2`, P(B) = `1/3` and P(A ? B) = `1/4`, then find

1) P(A / B)

2) P(B / A)

Question 4.

In a race, the probabilities of A and B winning the race are `1/3` and `1/6` respectively. Find the probability of neither of them winning the race.

Question 5.

A speaks the truth in 60% of the cases, while B is 40% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact?

Question 6.

From a lot of 6 items containing 2 defective items, a sample of 4 items are drawn at random. Let the random variable X denote the number of defective items in the sample.

If the sample is drawn without replacement, find :

1) The probability distribution of X

2) Mean of X

3) Variance of X

Question 7.

A card is drawn from a well-shuffled pack of playing cards. What is the probability that it is either a spade or an ace or both?

Question 8.

An urn contains 2 white and 2 black balls. A ball is drawn at random. If it is white, it is not replaced into the urn. Otherwise, it is replaced with another ball of the same colour. The process is repeated. Find the probability that the third ball is drawn is black.

Question 9.

Three persons A, B and C shoot to hit a target. If A hits the target four times in five trials, B hits it three times in four trials and C hits it two times in three trials, find the probability that:

1) Exactly two persons hit the target.

2) At least two persons hit the target.

3) None hit the target.

Question 10.

Box I contains two white and three black balls. Box II contains four white and one black balls and box III contains three white ·and four black balls. A dice having three red, two yellow and one green face, is thrown to select the box. If red face turns up, we pick up the box I, if a yellow face turns up we pick up box II, otherwise, we pick up box III. Then, we draw a ball from the selected box. If the ball is drawn is white, what is the probability that the dice had turned up with a red face?