Solutions - 5050-02

(1) Compute individual probabilities:

$P[A]=\frac{1}{2}$, $$ $P[B]=\frac{1}{2}$, $$ $P[C]=\frac{1}{2}$.

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(2) Verify pairwise independence:

$A\cap B$ occurs for $\{THT,HTH\}$, so $P[A\cap B]=\frac{1}{4}=P[A]P[B]$ $✓$

$B\cap C$ occurs for $\{TTH,HHT\}$, so $P[B\cap C]=\frac{1}{4}=P[B]P[C]$ $✓$

$A\cap C$ occurs for $\{HTT,THH\}$, so $P[A\cap C]=\frac{1}{4}=P[A]P[C]$ $✓$

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(3) Disprove mutual independence:

$A$, $B$, and $C$ cannot all occur simultaneously, so

$$P[A\cap B\cap C]=0\ne \frac{1}{8}=P[A]P[B]P[C]$$