Question

A coin tossed three times, where
$E$ is the event of head on third toss, $F$ is the event of heads on first two tosses. Then the value of $P\left(E/F\right)$ is:

• A. $\frac{1}{8}$
• B. $\frac{1}{6}$
• C. $\frac{1}{2}$
• D. $\frac{1}{4}$

Answer 1

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

Given: $E:$ Head on third toss.
$F:$ Heads on first two tosses
$\Rightarrow E=\{HHH,HTH,THH,TTH\}$ and $F=\{HHH,HHT\}$
$\Rightarrow E\cap F=\left\{HHH\right\}$
In this experiment, the sample space is of $8$ elements.
So, $P\left(E\right)=\frac{4}{8}=\frac{1}{2},P\left(F\right)=\frac{2}{8}=\frac{1}{4},P(E\cap F)=\frac{1}{8}$
Now we know that by definition of conditional probability
$P\left(E/F\right)=\frac{P(E\cap F)}{P\left(F\right)}$
$\Rightarrow P\left(E/F\right)=\frac{1/8}{1/4}=\frac{1}{2}$