Expert: Abe Mantell - 12/29/2007

Question
I am having difficulties understanding this problem and can't seem to get a correct answer, even though I have turned the assignment in 3 times. Please help if you can.
The following information, question and comments are word for word from the assignment.

Information: Separate the 12 face cards from a standard deck of 52 playing cards. Assume that the remaining cards have been shuffled. Select three cards from the pile of face cards.

Question: How many ways are there of selecting three of the same face cards (i.e. 3 jacks, 3 queens, 3 kings) from the pile?

Here is the feedback from my professor: In probability it is necessary to follow a solution that accounts for the total number of cards available, in this case 12 face cards. The same face card needs to be drawn each time. If a jack is drawn first, how many jacks are left in the deck for the second draw? The third?

This is all Greek to me. I can solve probability with marbles in a bag but the cards are throwing me for a loop. Can you please explain the steps and give the answer? I am taking a test on this stuff next week and am desperate.

Answer
Hello (again) Celeste,

Consider the number of ways of selecting 3 Jacks:
since there are 4 Jacks, there are C(4,3)=4 ways.
That is, there are 4 combinations.  Now consider the
total number of ways of selecting 3 cards out of 12:
there are C(12,3)=220 combinations.  Thus, the probability
of is 4/220 of selecting 3 Jacks.  Similarly, the prob.
of selecting 3 Queens is also 4/220, and the same for
3 Kings.  So, the total probability is:
4/220 + 4/220 + 4/220 = 12/220 = 3/55 = 0.05454...

Another way, as your instructor indicated, is:
The probabilty of selecting the first jack is 4/12
the second jack is 3/11, the third jack is 2/10
Thus, the probability of selecting three consecutive is
(4/12)(3/11)(2/10)=24/1320=4/220 (as I have above)
Then, the remaining probabilities work out the same.

See?