Question

Take 5 cards from a 52 card deck without replacement.

a) Use the multiplication principle reason out the number of ways to get 1 pair

and 3 other non-paired cards

               (Hint: Fill in the number of ways to choose: 1 card type, 1 matching card,

1 card that doesn’t match, 1 card that doesn’t match any of the previous 3, 1 card that doesn’t match any of the previous.

b) How many ways can you draw out a set of 5 cards?

               c) What’s the probability of exactly one pair?

               d) How many times would you expect to exactly one pair in 50 trials?

               e) Run 50 trails (shuffle each time if you’re using a real deck of cards)

                              How many times did you get exactly one pair?

                              Is this consistent with what’d you expect from part (d)?

Answer 1

a) We know there are 4 suits in a deck of cards and each suit contains 13 cards.

So at first, we can get 1 pair in    ways.

Now if we need 3 other non-paired cards, we have to select them from the other 3 suits and each one of them should be from a different suite.

So we can select the 3rd card in      ways and the 4th card in      and the last one in   .

Hence we can get 1 pair and 3 other non-paired cards in   .

b) We can draw a set of 5 cards in  

c) The probability of getting exactly one pair in a set of 5 cards is  

d) Let      be the number of times we get exactly one pair in a set of 5 cards in 50 trials.

So the expected number is   .

e) (After running a simulation)

I didn't get any pair in 50 trials.

Yes, this is consistent with what I expected from part (d).

If my answer is helpful for you, a like is very much appreciated.