Question

A fair coin is tossed a fixed number of times. If the probability of getting exactly $3$ heads equals the probability of getting exactly $5$ heads, then the probability of getting exactly one head is

• A. $1/64$
• B. $1/32$
• C. $1/16$
• D. $1/8$

Answer 1

The correct option is B $1/32$

Let the total number of trials be $n$.
The probability of getting exactly $3$ heads equals the probability of getting exactly $5$ heads,
${}^n{C}_3{\left(\frac{1}{2}\right)}^3{\left(\frac{1}{2}\right)}^{n-3}={}^n{C}_5{\left(\frac{1}{2}\right)}^5{\left(\frac{1}{2}\right)}^{n-5}$

$\Rightarrow n=8$
Therefore, the probability of getting exactly one head is
${}^8{C}_1\times {\left(\frac{1}{2}\right)}^8=\frac{1}{32}$