Consider 5 independent Bernoulli trials each | vedclass

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Question

(B) Let $X$ be the number of successes in $5$ independent Bernoulli trials. The probability of success is $p$ and the probability of failure is $q = 1

Vedclass · Accepted Answer

$[0, \frac{1}{2}]$

Vedclass · Answer

(B) Let $X$ be the number of successes in $5$ independent Bernoulli trials. The probability of success is $p$ and the probability of failure is $q = 1 - p$.The probability of at least one failure is given by $1 - P(	ext{no failure}) = 1 - P(X = 5)$.Since the trials are independent,$P(X = 5) = p^5$.According to the problem,the probability of at least one failure is at least $\frac{31}{32}$:$1 - p^5 \geq \frac{31}{32}$Rearranging the inequality:$1 - \frac{31}{32} \geq p^5$$\frac{1}{32} \geq p^5$Taking the fifth root on both sides:$p \leq (\frac{1}{32})^{1/5}$$p \leq \frac{1}{2}$Since $p$ is a probability,$p \geq 0$. Therefore,$p \in [0, \frac{1}{2}]$.

Consider $5$ independent Bernoulli trials each with probability of success $p$. If the probability of at least one failure is greater than or equal to $\frac{31}{32}$,then $p$ lies in the interval:

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