Question

4.2 theoretical probability. calculating theoretical probabilities (deck of cards). as shown above, a classic deck of playing cards is made up of 52 cards, 26 of which are black and the other 26 are red. each color is split into two suits of 13 cards each (clubs & spades are black, and hearts & diamonds are red). each suit is split into 13 ranks of cards (ace, 2 - 10, jack, queen, and king). if you select a card at random, what is the probability of getting... (a) ...a 5 of hearts? (b) ...a club or heart? (c) ...a number smaller than 9 (counting the ace as a 1)? question help: message instructor

Explanation:

Step1: Recall probability formula

The probability formula is $P(E)=\frac{n(E)}{n(S)}$, where $n(E)$ is the number of favorable outcomes and $n(S)$ is the total number of outcomes. Here, $n(S) = 52$.

Step2: Calculate probability for (a)

There is only 1 five - of - hearts in the deck. So $n(E)=1$. Then $P(\text{5 of Hearts})=\frac{1}{52}$.

Step3: Calculate probability for (b)

There are 13 clubs and 13 hearts. So $n(E)=13 + 13=26$. Then $P(\text{Club or Heart})=\frac{26}{52}=\frac{1}{2}$.

Step4: Calculate probability for (c)

The ranks smaller than 9 (counting ace as 1) are ace, 2, 3, 4, 5, 6, 7, 8. There are 8 ranks and 4 suits. So $n(E)=8\times4 = 32$. Then $P(\text{number}<9)=\frac{32}{52}=\frac{8}{13}$.

Answer:

(a) $\frac{1}{52}$
(b) $\frac{1}{2}$
(c) $\frac{8}{13}$