Embibe Experts Solutions for Chapter: Probability, Exercise 4: EXERCISE-4

Two numbers $x$ and $y$ are chosen at random from the set $\left\{1,2,3,4,\mathrm{.......3}n\right\}$. Find the probability that $\left(x^2-y^2\right)$ is divisible by $3$.

In a given race, the odds in favour of four horses $A,B,C$ & $D$ are $1:3,1:4,1:5$ and $1:6$ respectively. Assuming that a dead heat is impossible, find the chance that one of them wins the race.

$3$ students $\left[A,B,C\right]$ tackle a puzzle together and offers a solution upon which majority of the $3$ agrees. Probability of $A$ solving the puzzle correctly is $p$. Probability of $B$ solving the puzzle correctly is also $p$. $C$ is a dump student who randomly supports the solution of either $A$ or $B$. There is one more student $D$, whose probability of solving the puzzle correctly is once again, $p$. Out of the $3$ member team $\left[A,B,C\right]$ and one member team $\left\{D\right\}$, which one is more likely to solve the puzzle correctly.

Consider the following events for a family with children $A=\left\{\mathrm{of} \mathrm{both} \mathrm{sexes}\right\};B=\left\{\mathrm{at} \mathrm{most} \mathrm{one} \mathrm{boy}\right\}$

In which of the following (are/is) the events $A$and $B$ are independent.

$\left(a\right)$ If a family has $3$ children.

$\left(b\right)$ If a family has $2$ children

Assume that the birth of boy or a girl is equally likely mutually exclusive and exhaustive.

Each of the '$n$' passengers sitting in a bus may get down from it at the next stop with probability $p$. Moreover, at the next stop either no passenger or exactly one passenger boards the bus. The probability of no passenger boarding the bus at the next stop being $p_0$. Find the probability that when the bus continues on its way after the stop, there will again be '$n$' passengers in the bus.

An examination consists of $8$ questions in each of which the candidate must say which one of the $5$ alternatives is correct one. Assuming that the student has not prepared earlier chooses for each of the question any one of $5$ answers with equal probability.

$\left(a\right)$ Prove that the probability that he gets more than one correct answer is $\frac{\left(5^8-3\times 4^8\right)}{5^8}$.

$\left(b\right)$ Find the probability that he gets correct answers to six or more questions.

A purse contains $n$ coins of unknown value, a coin drawn at random is found to be a rupee, what is the chance that it is the only rupee in the purse? Assume all numbers of rupee coins in the purse to be equally likely.

A biased coin which comes up heads three time as often as tails is tossed. If it shows head, a chip is drawn from urn-I which contains $2$ white chips and $5$ red chips. If the coin comes up tail, a chip is drawn from urn-II which contains $7$ white and $4$ red chips. Given that a red chip was drawn, what is the probability that the coin came up head?