To find the probability of drawing 7 or more face cards (which includes Jacks, Queens, Kings, and Aces) from 8 random cards, we first need to recognize that in a standard deck of 52 cards, there are 16 face cards (4 each of Jacks, Queens, Kings, and Aces) and 36 non-face cards.
We need to calculate the probabilities of two cases:
Drawing exactly 7 face cards and 1 non-face card.
Drawing 8 face cards.
For both, we can use the hypergeometric distribution. The general formula for the hypergeometric probability is:
( N ) is the total number of items (cards in the deck, 52),
( K ) is the total number of items of one type (face cards, 16),
( n ) is the number of items to be drawn (8),
( k ) is the number of items of one type to be drawn.
For 7 face cards:
[ P(X = 7) = \frac{{\binom{16}{7} \binom{36}{1}}}{{\binom{52}{8}}} ]
For 8 face cards:
[ P(X = 8) = \frac{{\binom{16}{8} \binom{36}{0}}}{{\binom{52}{8}}} ]
We will calculate these probabilities to get the final answer.
The probability of drawing 7 or more face cards (Jacks, Queens, Kings, Aces) from 8 random playing cards is approximately 0.0564%, or about 1 in 1772. This is a rare event given the small proportion of face cards in a standard deck.
Put it in ChatGPT, just for fun…
To find the probability of drawing 7 or more face cards (which includes Jacks, Queens, Kings, and Aces) from 8 random cards, we first need to recognize that in a standard deck of 52 cards, there are 16 face cards (4 each of Jacks, Queens, Kings, and Aces) and 36 non-face cards.
We need to calculate the probabilities of two cases:
For both, we can use the hypergeometric distribution. The general formula for the hypergeometric probability is:
[ P(X = k) = \frac{{\binom{K}{k} \binom{N-K}{n-k}}}{{\binom{N}{n}}} ]
Where:
For 7 face cards: [ P(X = 7) = \frac{{\binom{16}{7} \binom{36}{1}}}{{\binom{52}{8}}} ]
For 8 face cards: [ P(X = 8) = \frac{{\binom{16}{8} \binom{36}{0}}}{{\binom{52}{8}}} ]
We will calculate these probabilities to get the final answer.
The probability of drawing 7 or more face cards (Jacks, Queens, Kings, Aces) from 8 random playing cards is approximately 0.0564%, or about 1 in 1772. This is a rare event given the small proportion of face cards in a standard deck.