Problem

A card is picked from a standard deck of 52 cards. Determine the odds against and the odds in favor of selecting a face card (king, queen, or jack).

The odds against selecting a face card are $\square \square$
(Type integers or decimals)
The odds in favor of selecting a face card are $\square, \square$
(Type integers or decimals)
(1) Time R

Answer

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Answer

Final Answer: The odds against selecting a face card are \(\boxed{10:3}\) and the odds in favor of selecting a face card are \(\boxed{3:10}\).

Steps

Step 1 :First, we need to determine the total number of face cards in a standard deck. A standard deck has 4 suits (hearts, diamonds, clubs, spades), and each suit has 3 face cards (king, queen, jack). So, there are \(4 \times 3 = 12\) face cards in total.

Step 2 :The odds against selecting a face card are calculated as the number of ways of not selecting a face card divided by the number of ways of selecting a face card. The number of ways of not selecting a face card is the total number of cards minus the number of face cards, which is \(52 - 12 = 40\). So, the odds against are \(40:12\).

Step 3 :The odds in favor of selecting a face card are calculated as the number of ways of selecting a face card divided by the number of ways of not selecting a face card. So, the odds in favor are \(12:40\).

Step 4 :However, these odds can be simplified by dividing both numbers by their greatest common divisor. The greatest common divisor of 40 and 12 is 4. So, the simplified odds against are \(\frac{40}{4}:\frac{12}{4} = 10:3\) and the simplified odds in favor are \(\frac{12}{4}:\frac{40}{4} = 3:10\).

Step 5 :Final Answer: The odds against selecting a face card are \(\boxed{10:3}\) and the odds in favor of selecting a face card are \(\boxed{3:10}\).

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