7.9 Exercise 1 — Contemporary Mathematics

April is playing a coin-flipping game with Ben. She will flip a coin 3 times. If the coin lands on heads more than tails, April wins; if it lands on tails more than heads, Ben wins. Let $A$ be the event “April wins,” $H$ be “first flip is heads,” and $T$ be “first flip is tails.” Compute $P \left(A\right)$ , $P \left(\right. A \left|\right. H \left.\right)$ , and $P \left(\right. A \left|\right. T \left.\right)$ .
You are about to draw 2 cards without replacement from a deck containing only these 10 cards: $\text{A} ♡$ , $\text{A} ♠$ , $\text{A} ♣$ , $\text{A} ♢$ , $\text{K} ♠$ , $\text{K} ♣$ , $\text{Q} ♡$ , $\text{Q} ♠$ , $\text{J} ♡$ , $\text{J} ♠$ . We’ll define the following events: $F$ is “both cards are the same rank,” $A$ is “first card is an ace,” and $K$ is “first card is a king.” Compute $P \left(\right. F \left|\right. A \left.\right)$ and $P \left(\right. F \left|\right. K \left.\right)$ .
Jim’s sock drawer contains 5 black socks and 3 blue socks. To avoid waking his partner, Jim doesn’t want to turn the lights on, so he puts on 2 socks at random. Let $M$ be the event “Jim’s 2 socks match,” let $K$ be the event “the sock on Jim’s left foot is black,” and let $L$ be the event “the sock on Jim’s left foot is blue.” Compute $P \left(M\right)$ , $P \left(\right. M \left|\right. K \left.\right)$ , and $P \left(\right. M \left|\right. L \left.\right)$ .

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