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Conditional Probability shark 1 min read

Q: If I flip 3 coins, what’s the probability that I get exactly 2 heads given that I get at least 1 tail?

A:

(1)   \begin{align*}P(\text{2 heads} | \geq \text{1 tail}) & = \frac{P(\text{2 heads} \cap \geq \text{1 tail})}{P(\geq \text{1 tail})} \\P(\geq \text{1 tail}) & = 1 - P(\text{all heads}) \\& = 1 - \frac{1}{8} \\& = \frac{7}{8} \\P(\text{2 heads} \cap \geq \text{1 tail}) & = C_3^1 (\frac{1}{8})^3 \\& = \frac{3}{8} \\P(\text{2 heads} | \geq \text{1 tail}) & = \frac{\frac{3}{8}}{\frac{7}{8}} \\& = \frac{3}{7}\end{align*}


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