Question

A coin is tossed three times, where (A) E: Head on the third toss, F: Heads on the first two tosses. Determine $\mathrm{P}\left(\mathrm{E}|\mathrm{F}\right)$.

Answer 1

Step 1:Finding the value of $P\left(E\right)$and $P\left(F\right)$

When a coin is tossed three times,

Therefore, $\mathrm{S}=\{\mathrm{HHH},\mathrm{HHT},\mathrm{HTH},\mathrm{THH},\mathrm{TTH},\mathrm{THT},\mathrm{HTT},\mathrm{TTT}\}$

E : Head On Third Toss

$\mathrm{E}=\{\mathrm{HHH},\mathrm{HTH},\mathrm{THH},\mathrm{TTH}\}$

Now, $\mathrm{P}\left(\mathrm{E}\right)=\frac{\mathrm{Favourable}\mathrm{outcome}}{\mathrm{Total}\mathrm{outcome}}$

$P\left(E\right)=\frac{4}{8}=\frac{1}{2}$

F : Head on the first two tosses.

$\mathrm{F}=\{\mathrm{HHH},\mathrm{HHT}\}$

Now, $\mathrm{P}\left(\mathrm{F}\right)=\frac{\mathrm{Favourable}\mathrm{outcome}}{\mathrm{Total}\mathrm{outcome}}$

$P\left(F\right)=\frac{2}{8}=\frac{1}{4}$

Step 2: Find the $\mathrm{P}\left(\mathrm{E}\cap \mathrm{F}\right)$

Now, we know that, to find $\mathrm{P}\left(\mathrm{E}|\mathrm{F}\right)$, we need to find the $\mathrm{P}\left(\mathrm{E}\cap \mathrm{F}\right)$.

So, $\left(\mathrm{E}\cap \mathrm{F}\right)=\left\{\mathrm{HHH}\right\}$

$\therefore \mathrm{P}\left(\mathrm{E}\cap \mathrm{F}\right)=\frac{1}{8}$.

We know that, $\mathrm{P}\left(\mathrm{E}|\mathrm{F}\right)=\frac{\mathrm{P}\left(\mathrm{E}\cap \mathrm{F}\right)}{\mathrm{P}\left(\mathrm{F}\right)}$

$$\begin{gathered} \mathrm{P}\left(\mathrm{E}|\mathrm{F}\right)=\frac{{\displaystyle \frac{1}{8}}}{{\displaystyle \frac{1}{4}}} \\ P\left(E|F\right)=\frac{4}{8}=\frac{1}{2} \end{gathered}$$

Hence, $\mathrm{P}\left(\mathrm{E}|\mathrm{F}\right)=\frac{1}{2}=0.5$.