Question

here is a table showing all 52 cards in a standard deck.
table with columns: color, suit, ace, two, three, four, five, six, seven, eight, nine, ten, jack, queen, king; rows: red-hearts, red-diamonds, black-spades, black-clubs with corresponding card symbols
suppose one card is drawn at random from a standard deck.
(a) find the odds in favor of drawing a spade.
1 : 3
(b) find the odds against drawing a four.
12 : 1

Explanation:

Part (a)

Step 1: Count spades and non - spades

In a standard deck, there are 13 spades (since each suit has 13 cards: Ace - King). The total number of non - spade cards is \(52 - 13=39\).
The odds in favor of an event is the ratio of the number of favorable outcomes to the number of unfavorable outcomes. For drawing a spade, the number of favorable outcomes (spades) is 13 and the number of unfavorable outcomes (non - spades) is 39.
We simplify the ratio \(\frac{13}{39}=\frac{1}{3}\), so the odds in favor of drawing a spade is \(1:3\).

Step 2: Final check

The formula for odds in favor of an event \(E\) is \(\text{Odds in favor of }E=\frac{\text{Number of favorable outcomes}}{\text{Number of unfavorable outcomes}}\). Here, favorable (spades) = 13, unfavorable (non - spades)=39, and \(\frac{13}{39} = \frac{1}{3}\) or \(1:3\).

Part (b)

Step 1: Count fours and non - fours

In a standard deck, there are 4 fours (one four in each suit: hearts, diamonds, spades, clubs). The number of non - four cards is \(52 - 4 = 48\).
The odds against an event is the ratio of the number of unfavorable outcomes to the number of favorable outcomes. For drawing a four, the number of unfavorable outcomes (non - fours) is 48 and the number of favorable outcomes (fours) is 4.
We simplify the ratio \(\frac{48}{4}=\frac{12}{1}\), so the odds against drawing a four is \(12:1\).

Step 2: Final check

The formula for odds against an event \(E\) is \(\text{Odds against }E=\frac{\text{Number of unfavorable outcomes}}{\text{Number of favorable outcomes}}\). Here, unfavorable (non - fours) = 48, favorable (fours)=4, and \(\frac{48}{4}=12\) or \(12:1\).

Answer:

(a) The odds in favor of drawing a spade is \(\boldsymbol{1:3}\).

(b) The odds against drawing a four is \(\boldsymbol{12:1}\).