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 in favor of drawing a spade.
(b) find the odds against drawing a four.

Explanation:

Part (a)

Step1: Determine number of spades and non - spades

A standard deck has 52 cards. The number of spade cards, \(n(\text{spade}) = 13\) (since there are 13 cards in each suit). The number of non - spade cards, \(n(\text{non - spade})=52 - 13=39\).

Step2: Calculate odds in favor

The formula for odds in favor of an event \(E\) is \(\text{Odds in favor of }E=\frac{n(E)}{n(\text{not }E)}\). For the event of drawing a spade, \(E=\text{spade}\), so odds in favor \(=\frac{13}{39}=\frac{1}{3}\), which can be written as \(1:3\) (or \(1\) to \(3\)).

Step1: Determine number of fours and non - fours

In a standard deck, there are 4 four - cards (one four in each suit). The number of non - four cards, \(n(\text{non - four}) = 52-4 = 48\).

Step2: Calculate odds against

The formula for odds against an event \(E\) is \(\text{Odds against }E=\frac{n(\text{not }E)}{n(E)}\). For the event of drawing a four, \(E = \text{four}\), so odds against \(=\frac{48}{4}=\frac{12}{1}\), which can be written as \(12:1\) (or \(12\) to \(1\)).

Answer:

\(1:3\) (or \(1\) to \(3\))

Part (b)