I get about 1/1772 but I’m not confident
((16 choose 7) * 36 + (16 choose 8)) / (52 choose 8)
Works out to about 1 chance in 1772, you are correct
36 is the number of non face cards in the deck – there are 16 face cards and 36 non face cards to choose from.
You might already know most of this, but just to break it down, the number of unordered selections of either 7 or 8 face cards is the sum of:
- (16 choose 7), the number of selections of 7 face cards out of the 16 available
- times 36, i.e. 52-16, the number of selections for the final, non-face card
- then plus (16 choose 8), the number of selections of 8 face cards
So the first term gives us the number of hands that we’re interested in, and then we divide it by the total number of hands possible (52 choose 8) to get probability.
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:
- 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:
[ P(X = k) = \frac{{\binom{K}{k} \binom{N-K}{n-k}}}{{\binom{N}{n}}} ]
Where:
- ( 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.
I tired “7 or 8 out of 8 face cards from deck of cards” on https://www.wolframalpha.com/ and it doesn’t look right, but it does seem to be in the right track.
