A person tossing a biased coin indefinitely w | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let the probability of getting a head be $p$. Then the probability of not getting a head is $1-p$. The person wins in an odd number of tosses if t

Vedclass · Accepted Answer

$\frac{32}{243}$

Vedclass · Answer

(D) Let the probability of getting a head be $p$. Then the probability of not getting a head is $1-p$. The person wins in an odd number of tosses if the first head appears on the $1^{st}, 3^{rd}, 5^{th}, \dots$ toss. This forms a geometric series: $p + (1-p)^2 p + (1-p)^4 p + \dots = 3/4$. Using the sum formula for an infinite geometric series $S = \frac{a}{1-r}$, where $a = p$ and $r = (1-p)^2$: $\frac{p}{1-(1-p)^2} = \frac{3}{4}$ $\frac{p}{1-(1-2p+p^2)} = \frac{3}{4}$ $\frac{p}{2p-p^2} = \frac{3}{4}$ $\frac{1}{2-p} = \frac{3}{4}$ $4 = 6 - 3p \implies 3p = 2 \implies p = 2/3$. If $5$ such coins are tossed at a time, the probability that a head appears on all $5$ coins is $p^5 = (2/3)^5 = 32/243$.

$A$ person tossing a biased coin indefinitely wins the game by getting head for the first time. The probability that he wins the game in an odd number of tosses is $3/4$. If $5$ such coins are tossed at a time, then the probability that head appears on all the coins is:

Similar Questions

It is observed that $25 \%$ of the cases related to child labour reported to the police station are solved. If $6$ new cases are reported,then the probability that at least $5$ of them will be solved is

$X$ denotes the number of times heads occur in $n$ tosses of a fair coin. If $P(X=4)$,$P(X=5)$,and $P(X=6)$ are in arithmetic progression,the largest value of $n$ is

In a game,a pair of dice is rolled $24$ times. If a person wins the game by not getting $6$ on both the dice in any one of the $24$ rolls,then the probability that a person wins the game is

If the mean and variance of a binomial distribution are $4$ and $2$ respectively,then the probability of $2$ successes of that binomial variate $X$ is:

Given that the probability of a man hitting a target with a gun is $\frac{1}{3}$. If he fires $8$ times,then the probability of his hitting the target at least twice is

Vedclass Test Series

Exam Paper Generator

Online Exam Module