# NCERT Class 12-Mathematics: Chapter – 13 Probability Part 7 (For CBSE, ICSE, IAS, NET, NRA 2022)

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**Question 8**:

Three events A, B and C have probabilities respectively. Given that , find the values of and .

**Answer**:

Here,

And

**Question 9**:

Let and be two independent events such that and . Describe in words of the events whose probabilities are:

(i)

(ii)

(iii)

(iv)

**Answer**:

(i) and occur

(ii) does not occur, but occurs

(iii) Either or , or both and occurs

(iv) Either or occurs, but not both

**Question 10**:

A discrete random variable X has the probability distribution given as below:

(i) Find the value of

(ii) Determine the mean of the distribution.

**Answer**:

We have

(i) we know that,

(ii) Mean of the distribution

**Question 11**:

Prove that

(i)

(ii)

**Answer**:

(i)

Hence proved.

(ii)

Hence proved.

**Question 12**:

If X is the number of tails in three tosses of a coin, determine the standard deviation of X.

**Answer**:

**Question 13**:

In a dice game, a player pays a stake of Re1 for each throw of a die. She receives ₹ 5 if the die shows a 3, ₹ 2 if the die shows a 1 or 6, and nothing otherwise. What is the player՚s expected profit per throw over a long series of throws?

**Answer**: