Odds against a gambler winning a bet that a card drawn from a pack of ordinary cards is a spade or an ace

Problem 5

A card is drawn from an ordinary pack and a gambler bets that it is a spade or an ace. What are the odds against his winning this bet?
Ans : 9 : 4

Solution

Experiment :

Drawing a card from a pack of 52 cards

Total Number of Possible Choices

= Number of ways in which one card can be drawn from the total 52

⇒ n = 52C1
=
52
1
= 52

Let

A : the event of the card drawn being a spade or an ace.

For Event A

Favorable
(Spades + Aces)
Unfavorable
(Others)
Total
Available 16 36 52
To Choose 1 0 1
Choices 16C136C052C1

Number of Favorable Choices

= Number of ways in which one card can be drawn from the total 16 favorable cards

⇒ mA = 16C1
=
16
1
= 16

Odds

Number of Unfavorable Choices

= Total Number of possible choices − Number of Favorable choices

⇒ mAc = n − mA
= 52 − 16
= 36

Odds against the gambler winning the bet

⇒ Odds against the card drawn being a spade or an ace.

⇒ Odds against Event A

= Number of Unfavorable Choices : Number of Favorable Choices

= mAc : mA

= 36 : 16

= 9 : 4