you are being dealt a 7-card hand from a standard deck of cards. what is the probability of being dealt 3 number cards (A-10) and 4 face cards (J, Q, K)?
There are a total of 12 number cards and 12 face cards in a standard deck of 52 cards.
The probability of being dealt a number card on the first draw is 24/52 (12/52 for a number card specifically). The probability of being dealt a number card on the second draw, given that a number card was already drawn, is 23/51 (11/51 for a number card specifically). The probability of being dealt a number card on the third draw, given that two number cards were already drawn, is 22/50 (10/50 for a number card specifically).
Similarly, the probability of being dealt a face card on the first draw is 12/52. The probability of being dealt a face card on the second draw, given that a face card was already drawn, is 11/51. The probability of being dealt a face card on the third draw, given that two face cards were already drawn, is 10/50.
Using the multiplicative principle, the probability of being dealt 3 number cards and 4 face cards is:
(12/52 x 11/51 x 10/50) x (12/49 x 11/48 x 10/47 x 9/46)
= 0.0172 or approximately 1.72%.