Question

Each of the persons $A$ and $B$ independently tosses three fair coins. The probability that both of them get the same number of heads is :

• A. $\frac{1}{8}$
• B. $1$
• C. $\frac{5}{8}$
• D. $\frac{5}{16}$

Answer 1

The correct option is D $\frac{5}{16}$

Let $A_i\to A$ gets $i$ heads.
$B_i\to B$ gets $i$ heads.
$P=P\left(\sum _{i=0}^3(A_i\cap B_i)\right)$
$=P\left(\sum _{i=0}^{3}P\left(A_i\right)P\left(B_i\right)\right)$
$=P\left(A_0\right)P\left(B_0\right)+P\left(A_1\right)\left(B_1\right)+P\left(A_2\right)\left(B_2\right)+P\left(A_3\right)\left(B_3\right)$
$=\frac{1}{8}\times \frac{1}{8}+\frac{3}{8}\times \frac{3}{8}+\frac{3}{8}\times \frac{3}{8}+\frac{1}{8}\times \frac{1}{8}$
$=\frac{5}{16}$