Conditional Probability and the Multiplication Rule · Example 1

(a) 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 $P\left(A\right)$, $P(A|H)$, and $P(A|T)$.
(b) 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 $P(F|A)$ and $P(F|K)$.
(c) 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 $P\left(M\right)$, $P(M|K)$, and $P(M|L)$.

OpenStax, Contemporary Mathematics · §7.9 Conditional Probability and the Multiplication Rule · Example 1