Embibe Experts Solutions for Chapter: Probability, Exercise 1: Manipur Board-2018

A family has two children. Find the probability that both the children are boys if it is known that at least one of the children is a boy.

A girl throws a dice. If she gets a $5$ or $6$, she tosses a coin three times and notes the number of heads. If she gets $1, 2, 3$ or $4$ she tosses a coin two times and notes the number of heads obtained. If she obtained exactly two heads, what is the probability that she throws $1, 2, 3$ or $4$ with the dice?

Mean and variance of a binomial distribution are $12$ and $3$ respectively. Then the number of trials is :

Two cards are drawn simultaneously from a well-shuffled pack of $52$ cards. Find the probability distribution of the number of aces.

State and prove Baye's Theorem.

What is meant by a random variable? Define its probability distribution, mean and variance.

In a bolt factory, machines $A, B$ and $C$ manufacture respectively $60\mathrm{\%}, 25\mathrm{\%}$ and $15\%$ of total bolts respectively. Of the total of their output $1\mathrm{\%}, 2\% \mathrm{and} 1\mathrm{\%}$ are defective bolts. A bolt is drawn at random from the total production and found to be defective. What is the probability that is manufactured by the machine $B$?  [Write the answer in fraction.]

A box contains $10$ items out of which $2$ are defective. A man selects $3$ items at random. Find the expected number of defective items he has drawn. [Write the answer in fraction.]