Question

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

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Solution

## Step 1:Finding the value of $P\left(E\right)$and $P\left(F\right)$When a coin is tossed three times,Therefore, $\mathrm{S}=\left\{\mathrm{HHH},\mathrm{HHT},\mathrm{HTH},\mathrm{THH},\mathrm{TTH},\mathrm{THT},\mathrm{HTT},\mathrm{TTT}\right\}$E : Head On Third Toss$\mathrm{E}=\left\{\mathrm{HHH},\mathrm{HTH},\mathrm{THH},\mathrm{TTH}\right\}$Now, $\mathrm{P}\left(\mathrm{E}\right)=\frac{\mathrm{Favourable}\mathrm{outcome}}{\mathrm{Total}\mathrm{outcome}}$$P\left(E\right)=\frac{4}{8}=\frac{1}{2}$F : Head on the first two tosses.$\mathrm{F}=\left\{\mathrm{HHH},\mathrm{HHT}\right\}$Now, $\mathrm{P}\left(\mathrm{F}\right)=\frac{\mathrm{Favourable}\mathrm{outcome}}{\mathrm{Total}\mathrm{outcome}}$$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\}$$\therefore \mathrm{P}\left(\mathrm{E}\cap \mathrm{F}\right)=\frac{1}{8}$.We know that, $\mathrm{P}\left(\mathrm{E}|\mathrm{F}\right)=\frac{\mathrm{P}\left(\mathrm{E}\cap \mathrm{F}\right)}{\mathrm{P}\left(\mathrm{F}\right)}$$\mathrm{P}\left(\mathrm{E}|\mathrm{F}\right)=\frac{\frac{1}{8}}{\frac{1}{4}}\phantom{\rule{0ex}{0ex}}P\left(E|F\right)=\frac{4}{8}=\frac{1}{2}$Hence, $\mathrm{P}\left(\mathrm{E}|\mathrm{F}\right)=\frac{1}{2}=0.5$.

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