# NCERT Class 12-Mathematics: Chapter – 13 Probability Part 9 (For CBSE, ICSE, IAS, NET, NRA 2023)

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**Question 24**:

Consider the probability distribution of a random variable X:

Calculate (i)

(ii) Variance of X.

**Answer**:

(i)

(ii)

**Question 25**:

The probability distribution of a random variable X is given below:

(i) Determine the value of .

(ii) Determine and

(iii) Find .

**Answer**:

(i)

(ii)

(iii)

**Question 26**:

For the following probability distribution determine standard deviation of the random variable X.

**Answer**:

**Question 27**:

A biased die is such that and other scores being equally likely. The die is tossed twice. If X is the ‘number of fours seen’ , find the variance of the random variable X.

**Answer**:

Since,

On tossing two die,

Also,

So,

Thus, we get following table

**Question 28**:

A die is thrown three times. Let be ‘the number of twos seen’ . Find the expectation of .

**Answer**:

We have,

So, on throwing a die three times, we will have .

We know that,

**Question 29**:

Two biased dice are thrown together. For the first die , the other scores being equally likely while for the second die, and the other scores are equally likely. Find the probability distribution of ‘the number of ones seen’ .

**Answer**: