Question

A coin is tossed 5 times. The probability that tail appears an odd number of times, is

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{2}{5}$
• D. $\frac{1}{5}$

Answer 1

The correct option is A $\frac{1}{2}$

We have,

$p=$ Probability of tail appear $=\frac{1}{2}$

$q=$ Probability of tail not appear $=1-p=\frac{1}{2}$

Now, $P$ (Tail appear odd number of times)

$=P(X=1)+P(X=3)+P(X=5)$

$=C_1^5\left(\frac{1}{2}\right){\left(\frac{1}{2}\right)}^4+$$C_3^5{\left(\frac{1}{2}\right)}^3{\left(\frac{1}{2}\right)}^2+$$C_5^5{\left(\frac{1}{2}\right)}^5$ $\because P(X=r)=C_r^n{p}^r{q}^{n-r}$

$=\frac{5}{32}+\frac{10}{32}+\frac{1}{32}$

$=\frac{16}{32}=\frac{1}{2}$

Hence, A is the correct option.