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 9 of diamonds? (b) ...a spade or diamond? (c) ...a number smaller than 10 (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 elements in the event set and $n(S)$ is the number of elements in the sample - space. Here, $n(S) = 52$ (total number of cards in a deck).

Step2: Calculate probability for (a)

There is only 1 nine - of - diamonds in the deck. So, $n(E)=1$. Then $P(\text{9 of Diamonds})=\frac{1}{52}$.

Step3: Calculate probability for (b)

There are 13 spades and 13 diamonds in the deck. So, $n(E)=13 + 13=26$. Then $P(\text{Spade or Diamond})=\frac{26}{52}=\frac{1}{2}$.

Step4: Calculate probability for (c)

In each suit, there are 9 cards with a number smaller than 10 (ace as 1, 2 - 9). Since there are 4 suits, $n(E)=9\times4 = 36$. Then $P(\text{number}<10)=\frac{36}{52}=\frac{9}{13}$.

Answer:

(a) $\frac{1}{52}$
(b) $\frac{1}{2}$
(c) $\frac{9}{13}$