Questions: Suppose one card is drawn at random from a standard deck. (a) Find the odds against drawing a red card. (b) Find the odds in favor of drawing a jack of spades.

Solution Steps

To solve these problems, we need to understand the composition of a standard deck of cards and use basic probability concepts.

(a) To find the odds against drawing a red card, we first determine the number of red cards in the deck and the total number of cards. The odds against an event are calculated as the ratio of the number of unfavorable outcomes to the number of favorable outcomes.

(b) To find the odds in favor of drawing a jack of spades, we identify the number of favorable outcomes (drawing the jack of spades) and the total number of cards. The odds in favor of an event are calculated as the ratio of the number of favorable outcomes to the number of unfavorable outcomes.

A standard deck of cards contains a total of 52 cards.

• There are 26 red cards in a standard deck (13 Hearts and 13 Diamonds).
• The odds against drawing a red card are calculated as the ratio of the number of non-red cards to the number of red cards: \[ \text{Odds against red} = \frac{\text{Total cards} - \text{Red cards}}{\text{Red cards}} = \frac{52 - 26}{26} = 1.0 \]

• There is only 1 jack of spades in the deck.
• The odds in favor of drawing the jack of spades are calculated as the ratio of the number of favorable outcomes to the number of unfavorable outcomes: \[ \text{Odds in favor of jack of spades} = \frac{1}{52 - 1} = 0.01961 \]

Final Answer