Question

5 cards are drawn at random from a standard deck.

Find the probability that all the cards are hearts.


Find the probability that all the cards are face cards.

Note: Face cards are kings, queens, and jacks.

Find the probability that all the cards are even.

(Consider aces to be 1, jacks to be 11, queens to be 12, and kings to be 13)

Answers can be entered as unreduced fractions.
Submit QuestionQuestion 5

Answer 1

1. The probability that all 5 cards drawn are hearts is
`\( (13)/(52) * (12)/(51) * (11)/(50) * (10)/(49) * (9)/(48) \)`.

2. The probability that all 5 cards drawn are face cards is
`\( (12)/(52) * (11)/(51) * (10)/(50) * (4)/(49) * (4)/(48) \).`

3. The probability that all 5 cards drawn are even is
`\( (18)/(52) * (16)/(51) * (14)/(50) * (10)/(49) * (8)/(48) \).`

Step-by-step explanation:

1. To find the probability that all 5 cards drawn are hearts, we consider that there are 13 hearts in a standard deck of 52 cards. The probability of drawing a heart on the first draw is
`\( (13)/(52) \)`. As cards are drawn without replacement, the probabilities for the subsequent draws are adjusted accordingly. The final probability is calculated by multiplying these individual probabilities.

2. For the probability that all 5 cards drawn are face cards, we note that there are 3 face cards in each suit, making a total of 12 face cards in a standard deck. The calculation involves the probability of drawing a face card on each draw, considering the reduced number of face cards after each draw.

3. To determine the probability that all 5 cards drawn are even, we account for the fact that there are 9 even-numbered cards (2, 4, 6, 8, 10, 12, 14, 16, 18) in each suit. The probability for each draw is calculated, adjusting for the reduced number of even cards in subsequent draws.

These probabilities are calculated using the fundamental principle of probability and the concept of conditional probability when drawing without replacement from a finite deck of cards.

Answer 2

Step-by-step explanation:

Probability of drawing all hearts:

There are 13 hearts in a standard deck of 52 cards. So, the probability of drawing the first heart is 13/52. Once the first heart is drawn, there are 12 hearts left out of 51 cards, so the probability of drawing a second heart is 12/51. Continuing this process, the probability of drawing all 5 hearts is:

(13/52) * (12/51) * (11/50) * (10/49) * (9/48) = 0.000495 or 1/2024

Therefore, the probability of drawing all 5 cards as hearts is 1/2024.

Probability of drawing all face cards:

There are 12 face cards in a standard deck (4 kings + 4 queens + 4 jacks), so the probability of drawing the first face card is 12/52. Once the first face card is drawn, there are 11 face cards left out of 51 cards, so the probability of drawing a second face card is 11/51. Continuing this process, the probability of drawing all 5 face cards is:

(12/52) * (11/51) * (10/50) * (9/49) * (8/48) = 0.000009 or 1/110,583

Therefore, the probability of drawing all 5 cards as face cards is 1/110,583.

Probability of drawing all even cards:

There are 20 even cards in a standard deck (2, 4, 6, 8, 10, 12), so the probability of drawing the first even card is 20/52. Once the first even card is drawn, there are 19 even cards left out of 51 cards, so the probability of drawing a second even card is 19/51. Continuing this process, the probability of drawing all 5 even cards is:

(20/52) * (19/51) * (18/50) * (17/49) * (16/48) = 0.000058 or 1/17,296

Therefore, the probability of drawing all 5 cards as even cards is 1/17,296.