At least how many times must a fair coin be t | vedclass

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Question

(D) Let $n$ be the number of times the coin is tossed. Let $X$ be the number of heads obtained. $X$ follows a binomial distribution with parameters $n

Vedclass · Accepted Answer

$3$

Vedclass · Answer

(D) Let $n$ be the number of times the coin is tossed. Let $X$ be the number of heads obtained. $X$ follows a binomial distribution with parameters $n$ and $p = \frac{1}{2}$.We are given that $P(X \ge 1) \ge 0.8$.Using the complement rule,$P(X \ge 1) = 1 - P(X = 0)$.So,$1 - P(X = 0) \ge 0.8 \Rightarrow P(X = 0) \le 0.2$.Since $P(X = 0) = \binom{n}{0} \left(\frac{1}{2}\right)^0 \left(1 - \frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^n$,we have $\left(\frac{1}{2}\right)^n \le 0.2$.$\Rightarrow \frac{1}{2^n} \le \frac{1}{5} \Rightarrow 2^n \ge 5$.For $n = 1$,$2^1 = 2 For $n = 2$,$2^2 = 4 For $n = 3$,$2^3 = 8 \ge 5$.Thus,the minimum number of tosses required is $3$.

At least how many times must a fair coin be tossed so that the probability of getting at least one head is at least $0.8$?

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