Odds against card selection

Question

find the odds against selecting a face card or a spade when a card is drawn from a standard deck of cards

Answer 1

"Odds in favor" is defined as the number of successes divided by the number of failures while "odds against" is defined as the number of failures divided by the number of successes. Obviously, we want the latter.

In this case, since a face card is Jack, Queen, or King, and there are 4 different suits, there are 12 face cards. We also know that there are 13 spades, HOWEVER, 3 of those spades are already included in the face cards we have counted. So the number of cards that are face cards OR spades is 12 + (13 - 3) = 22. That makes 30 cards that are NOT face cards or spades (52 total cards - 22 that are face cards or spades)

The number of failures is 30 and the number of successes is 22 the the odds against is 30:22 or 15:11

Answer 2

Assuming a standard deck of cards includes 52 cards with no Jokers.

There are 13 spade cards from A to K

Three of the spades are face cards (J Q K)

There are 12 face cards

Three for each suit.

Since we want to avoid double counting the J Q K of spades there are

13 spades and 9 face cards that aren't spades.

This is 22 total cards.

The probability is 22/52 or 11/26 of drawing one of the cards so the odds that this doesn't happen is one minus that.

1-11/26 = 15/26 chance to not draw a face or spades.

This is also a 57.7% chance if percentage is needed.

Answer 3

The probability of selecting a face card or a spade = P(face card) + P(spade) - P(face card and space) = 12/52 + 13/52 - 3/52 = 22/52 = 11/26. So probability of NOT selecting a face card or a space = 15/26, so odds are 15:11 against selecting a face card or a spade.