Question

If you are dealt 3 cards from a shuffled deck of 52​ cards, find the probability of getting one queen and two kings.

Answer 1

Answer:

that's the number of ways you can get 2 kings from 4 kings times the number of ways you can get 1 queen out of 4 queens divided by the number of ways you can get any 3 cards out of 52 cards.

4c3 = 4! / (2! * 2!) = 6

4c1 = 4! / (1! * 3!) = 4

52c3 = 52! / (3! * 49!) = 22100

the probability is therefore 24 / 22100 = .0010859729.

you can do it another way.

to get 2 kings and one queen out of deck of 52 cards give you 3 possibilities.

they are KKQ, KQK, and QKK.

that is draws 1, 2 and 3 in that order.

you can draw a king on the first draw and a king on the second draw and a queen on the third draw, or you can draw a kind on the first draw and a queen on the second draw and a king on the third draw, or you can draw a queen on the first draw and a king on the second draw and a king on the third draw.

Step-by-step explanation:

this is all without replacement, so the probability changes each draw.

for KKQ, the probaiblity is 4/52 * 3/51 * 4/50.

for KQK, the probability is 4/52 * 4/51 * 3/50.

for QKK, the probability is 4/52 * 4/51 * 3/50.

each oone of these probabilties is the same, so the total probability would be 3 * (4 * 3 * 4) / (52 * 51 * 50) = = .0010859729, the same as we got using the other method.

i believe that's your answer.

note that ncx is equal to n! / (x! * (n-x)!).

when n = 4 and x = 2, that becomes 4! / (2! * 2!).

when n = 4 and x = 1, that becomes 4! / (1! * 3!).

when n = 52 and x = 3, that becomes 52! / (3! * 49!).

ncx can also be written as c(n,x).

they mean the same thing.