# NCERT Class 12-Mathematics: Chapter – 13 Probability Part 6 (For CBSE, ICSE, IAS, NET, NRA 2023)

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## 13.3 EXERCISE

### Short Answer (S. A)

**Question 1**:

For a loaded die, the probabilities of outcomes are given as under: .

The die is thrown two times. Let A and B be the events, ‘same number each time’ , and ‘a total score is or more’ , respectively. Determine whether or not A and B are independent.

**Answer**:

A and B are Independents events.

**Question 2**:

Refer to Exercise 1 above. If the die were fair, determine whether or not the events A and B are independent.

**Answer**:

A and B are not Independent events

**Question 3**:

The probability that at least one of the two events A and B occurs is . If A and B occur simultaneously with probability , evaluate .

**Answer**:

We know that, denotes the occurrence of at least one of A and B and denotes the occurrence of both A and B, simultaneously.

Thus,

**Question 4**:

A bag contains 5 red marbles and 3 black marbles. Three marbles are drawn one by one without replacement. What is the probability that at least one of the three marbles drawn be black, if the first marble is red?

**Answer**:

Let

For at least one of the three marbles drawn be black, if the first marble is red, then the following three conditions will be followed.

(i) Second ball is black and third is red

(ii) Second ball is black and third is also black

(iii) Second ball is red and third is black

And

**Question 5**:

Two dice are thrown together and the total score is noted. The events E, F and G are ‘a total of 4’ , ‘a total of 9 or more’ , and ‘a total divisible by 5’ , respectively. Calculate , and and decide which pairs of events, if any, are independent.

**Answer**:

no pair is independent

**Question 6**:

Explain why the experiment of tossing a coin three times is said to have binomial distribution.

**Answer**:

We know that, a random variable X taking value is said to have a binomial distribution with parameters and , if its probability distribution is given by

Where

And

Similarly, in an experiment of tossing a coin three times, we have and random variable can take value with

So, we see that in the experiment of tossing coin three, we have random variable which can take values with parameters

Therefore, it is said to have a binomial distribution.

**Question 7**:

A and B are two events such that Find:

(i)

(ii)

(iii)

(iv)

**Answer**:

(ii)

(iii)

(iv)