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 club.
(b) find the odds in favor of drawing a six.

Explanation:

Step1: Calculate number of non - clubs

There are 52 cards in a deck and 13 clubs. So number of non - clubs is $52 - 13=39$.

Step2: Find odds against drawing a club

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

Step3: Calculate number of sixes

There are 4 sixes in a deck (one in each suit).

Step4: Calculate number of non - sixes

Number of non - sixes is $52 - 4 = 48$.

Step5: Find odds in favor of drawing a six

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

Answer:

(a) 3 to 1
(b) 1 to 12