What is Probability distribution?

Example 9.4

Roll a fair die: $\Omega = \{1, 2, 3, 4, 5, 6\}$, each outcome having probability $\frac16$. The event $A$: “the result is even” is $A = \{2, 4, 6\}$, so $\P(A) = \frac36 = \frac12$. The event $B$: “the result is at least $5$” is $B = \{5, 6\}$, so $\P(B) = \frac26 = \frac13$.

Example 9.7

Draw one card from a standard $52$-card deck. Let $A$: “the card is a heart” ($13$ cards) and $B$: “the card is a king” ($4$ cards). Then $A \cap B$ is “the king of hearts” ($1$ card), and

The complement rule is often the fastest route: the probability that the card is not a heart is $1 - \frac{13}{52} = \frac34$.

Example 9.9

An urn contains $3$ red balls and $2$ blue balls. Draw one ball, put it back, draw again. Each draw gives red (R) with probability $\frac35$ and blue (B) with probability $\frac25$.

The probability of getting two balls of the same color is $\P(\text{RR}) + \P(\text{BB}) = \frac{9}{25} + \frac{4}{25} = \frac{13}{25}$; the probability of at least one blue is $1 - \P(\text{RR}) = 1 - \frac{9}{25} = \frac{16}{25}$ (complement rule).