Question

In a simultaneous toss of n coins, find the probability that no head occurs.

• A. $\frac{1}{2n}$
• B. $\frac{1}{n}$
• C. $\frac{1}{2^n}$
• D. $\frac{1}{n^2}$

Answer 1

The correct option is C

$\frac{1}{2^n}$

Let us take a toss of 1 coin. The possible outcomes are 2: (H) ,(T) =$2^1$
In a toss of 2 coins, the possible outcomes are 4: (H, H), (H, T), (T, H), (T, T) = $2^2$
In a toss of 3 coins , the possible outcomes are 8 - (H, H, H), (H, T, H), (H, H, T), (H, T, T) , (T, H, H), (T, H, T), (T, T, H), (T, T, T) = $2^3$
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.
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In case of a toss of n coins, the total number of outcomes is $2^n$.

There is only 1 case in which no head occurs as can be seen from the above cases.(the case where all the coins show T)

$\therefore$ probability that no head occurs = $\frac{1}{2^n}$