A card is drawn from an ordinary pack of playing cards and a person bets that it is a spade or a king.What are odds against his winning this bet?
To calculate the odds against winning a bet, we need to first determine the number of unsuccessful outcomes (losing outcomes) and divide it by the number of successful outcomes.
In an ordinary pack of playing cards, there are 52 cards. The person is betting that the drawn card will be either a spade or a king. Let's calculate the number of successful outcomes and losing outcomes:
Number of Successful Outcomes:
- There are 13 spades in a deck of cards.
- There are 4 kings in a deck of cards (one king for each suit).
- However, we need to subtract one king because it is already counted as a spade (since it is both a king and a spade).
Therefore, the number of successful outcomes is 13 + 4 - 1 = 16.
Number of Unsuccessful Outcomes:
- The remaining 36 cards in the deck (52 total cards - 16 successful outcomes).
Now that we have the number of successful and unsuccessful outcomes, we can calculate the odds against winning this bet:
Odds against = Number of Unsuccessful Outcomes / Number of Successful Outcomes
Odds against = 36 / 16 = 2.25
Therefore, the odds against winning this bet are 2.25 to 1 (or 9/4 if you prefer fractions).
P(spade) = 1/4
P(king) = 1/13
P(spade v king) = P(spade) + P(king) - P(spade ^ king)
= 1/4 + 1/13 - 1/4 * 1/13
= (13+4-1)/(4*13)
= 16/52
= 4/13