Question

A deck of playing cards has four suits, with thirteen cards in each suit consisting of the numbers 2 through 10, a jack, a queen, a king, and an ace. The four suits are hearts, diamonds, spades, and clubs. A hand of five cards will be chosen at random.

Which statements are true? Check all that apply.

The total possible outcomes can be found using 52C5.
The total possible outcomes can be found using 52P5.
The probability of choosing two diamonds and three hearts is 0.089.
The probability of choosing five spades is roughly 0.05
The probability of choosing five clubs is roughly 0.0005.

Answer 1

True:
- The total possible outcomes can be found using 52C5.
(A 5-card hand is an unordered selection, so use combinations: C(52,5) = 2,598,960.)

False:
- The total possible outcomes can be found using 52P5.
(52P5 counts ordered sequences; order does not matter for a hand.)

- The probability of choosing two diamonds and three hearts is 0.089.
Correct probability = C(13,2)·C(13,3) / C(52,5) = (78·286)/2,598,960 = 22,308/2,598,960 ≈ 0.00858 (≈0.0086), not 0.089.

- The probability of choosing five spades is roughly 0.05.
Correct probability = C(13,5)/C(52,5) = 1,287/2,598,960 ≈ 0.000495 (≈0.0005), not 0.05.

True:
- The probability of choosing five clubs is roughly 0.0005.
Same calculation as for five spades: ≈0.000495 ≈ 0.0005.