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The Least Number of Times a Fair Coin Must Be Tossed So that the Probability of Getting at Least One Head is at Least 0.8, is (A) 7 (B) 6 (C) 5 (D) 3 - Mathematics

Question

The least number of times a fair coin must be tossed so that the probability of getting at least one head is at least 0.8, is

• 7

• 6

• 5

• 3

Solution

3

Let X denote the number of coins.
Then, X follows a binomial distribution with

$p = \frac{1}{2} , q = \frac{1}{2}$
$\text{ It is given that } P(X \geq 1) \geq 0 . 8$
$\Rightarrow 1 - P(X = 0) \geq 0 . 8$
$\Rightarrow P(X = 0) \leq 1 - 0 . 8$
$\Rightarrow P(X = 0) = 0 . 2$
$\Rightarrow \frac{1}{2^n} \leq 0 . 2$
$\Rightarrow 2^n \geq \frac{1}{0 . 2}$
$\Rightarrow 2^n \geq 5$
$\text{ This is possible when n } \geq 3$
$\text{ So, the least value of n is } 3 .$

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