Question

here is a table showing all 52 cards in a standard deck.
color suit ace two three four five six seven eight nine ten jack queen king
red hearts a♥ 2♥ 3♥ 4♥ 5♥ 6♥ 7♥ 8♥ 9♥ 10♥ j♥ q♥ k♥
red diamonds a♦ 2♦ 3♦ 4♦ 5♦ 6♦ 7♦ 8♦ 9♦ 10♦ j♦ q♦ k♦
black spades a♠ 2♠ 3♠ 4♠ 5♠ 6♠ 7♠ 8♠ 9♠ 10♠ j♠ q♠ k♠
black clubs a♣ 2♣ 3♣ 4♣ 5♣ 6♣ 7♣ 8♣ 9♣ 10♣ j♣ q♣ k♣
suppose one card is drawn at random from a standard deck.
(a) find the odds against drawing a spade.
(b) find the odds in favor of drawing a two.

Explanation:

Step1: Recall total number of cards

There are 52 cards in a standard deck.

Step2: Find number of spades and non - spades

There are 13 spades in a deck. So, the number of non - spades is $52 - 13=39$.

Step3: Calculate odds against drawing a spade

The odds against an event is the ratio of the number of unfavorable outcomes to the number of favorable outcomes. So, the odds against drawing a spade is $\frac{39}{13}=\frac{3}{1}$, or 3:1.

Step4: Find number of twos and non - twos

There are 4 twos in a deck. So, the number of non - twos is $52 - 4 = 48$.

Step5: Calculate odds in favor of drawing a two

The odds in favor of an event is the ratio of the number of favorable outcomes to the number of unfavorable outcomes. So, the odds in favor of drawing a two is $\frac{4}{48}=\frac{1}{12}$, or 1:12.

Answer:

(a) 3:1
(b) 1:12