Embibe Experts Solutions for Chapter: Probability, Exercise 1: Exercise

Suppose a girl throws a die. If she gets $1$ or $2$, she tosses a coin three times and notes the number of tails. If she gets $3,4,5,$ or $6$, she tosses a
coin once and notes whether a ‘head’ or ‘tail’ is obtained. If she obtained exactly one ‘tail’, what is the probability that she threw $3,4,5,$ or $6$ with the die ?

Let $E$ be an event in sample space and $E^{\text{'}}$ be its complement. If $P\left(E\right)=\frac{3}{4}$. Find $P\left(E\cup E^{\text{'}}\right)$.

A test detects a disease correctly $80\%$ of time. But it fails to detect it correctly for $2\%$ of the time. A person is chosen at random from a large population of which $0.6\%$ have the disease. He underwent the test and was reported to have the disease. What is the probability that the person actually has the disease$?$

In a certain college, $4\%$ of boys and $1\%$ of girls are taller than $1.75$ metres. Furthermore, $60\%$ of the students in the college are girls. A student selected at random from the college is found to be taller than $1.75$ metres. Find the probability that the selected student is a girl.

An examinee has to answer a question on a multiple choice type test. The answer to this question is either known to him or else he simply guesses at it. The probability that he knows the answer is $\frac{4}{5}$. He is correct in $25\%$ cases when he guesses at the answer. Find the probability that he knows the answer given that he answered it correctly.

$A$ and $B$ toss a coin alternately until one of them gets a head first time wins and wins the toss. If $A$ begins, find the respective probabilities for $A$ and $B$ to win the toss.

If the mean and variance of a binomial distribution are $8$ and $4$ respectively then P(X $=1$) is $\frac{1}{k}$, $k=$ ____

If $A \text{and} B$ are two independent events with $P\left(A\right)=\frac{1}{3}\text{ and }P\left(B\right)=\frac{1}{4}$, then $P\left(B^{\text{'}}\mid A\right)$ is equal to