Consider 5 independent Bernoulli trials,each | vedclass

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(A) Let $X$ be the number of successes in $5$ independent Bernoulli trials. $X$ follows a binomial distribution $B(n, p)$ with $n = 5$. The probabilit

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$[0, 1/2]$

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(A) Let $X$ be the number of successes in $5$ independent Bernoulli trials. $X$ follows a binomial distribution $B(n, p)$ with $n = 5$. The probability of success is $p$ and the probability of failure is $q = 1 - p$. The probability of at least one failure is $P(X Since $P(X = 5) = p^5$,the probability of at least one failure is $1 - p^5$. We are given that $1 - p^5 \ge 31/32$. Rearranging the inequality: $1 - 31/32 \ge p^5$,which simplifies to $1/32 \ge p^5$. Taking the fifth root on both sides: $(1/2)^5 \ge p^5$,which implies $p \le 1/2$. Since $p$ is a probability,$p \ge 0$. Thus,$p \in [0, 1/2]$.

Consider $5$ independent Bernoulli trials,each with a probability of success $p$. If the probability of at least one failure is greater than or equal to $31/32$,then in which interval does $p$ lie?

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