Collage Probability

Question

Five cards are drawn randomly from a pack of 52 playing cards .

(a) Find the probability that at most 2 Kings are drawn given that at least 1 King is drawn.

(b) Find the probability that 3 kings and 2 Queens are drawn given that at least 3 cards drawn are of the same number or the same letter.

Thanks for your help.

Simon YAU

Answer 1

(a) Find the probability that at most 2 Kings are drawn given that at least 1 King is drawn.

Let event B be the event "at least one king is drawn"

And let A be the event "at most 2 kings are drawn."

We want to calculate Pr(at most 2 kings are drawn given at least 1 king is drawn) = Pr(A | B) = #(A∩B) / #(B).

Where #(B) is the number of ways event B, "at least one king is drawn," can happen, and #(A∩B) is the number of ways event A∩B, "at most 2 kings are drawn and at least one king is drawn," can happen.

First figuring out #(B):

It helps to phrase B as "1 king is drawn or 2 kings are drawn or 3 kings are drawn or 4 kings are drawn." And phrase that as "1 king is drawn from 4 kings and the other 4 cards are drawn from the remaining 48, or 2 kings are drawn from 4 kings and the other 3 cards are drawn from the remaining 48, or 3 kings are drawn from 4 kings and the other 2 cards are drawn from the remain 48, or 4 kings are drawn from 4 kings and the 1 other card is drawn from the remaining 48."

Even though its a very long way of phrasing it, you can now see all the subcombinations you have to go through in order to count every single possible combination of at least one king being drawn.

#(B) = number of ways 1 king is drawn from 4 kings times the number of ways the other 4 cards are drawn from the remaining 48, plus the number of ways 2 kings are drawn from 4 kings times the number of ways the other 3 cards are drawn from the remaining 48, plus the number of ways 3 kings are drawn from 4 kings times the number of ways the other 2 cards are drawn from the remaining 48, plus number of ways 4 kings are drawn from 4 kings times the number of ways the 1 other card is drawn from the remaining 48.

To put more succinctly in mathematical notation, we have

#(B) = C(4,1)*C(48,4) + C(4,2)*C(48,3) + C(4,3)*C(48,2) + C(4,4)*C(48,1) = (4)*(194580) + (6)*(17296) + (4)*(1128) + (1)*(48) = 886656.

Where C(n,k) is n choose k.

Now figuring out #(A∩B). i

A∩B ≡ "at most 2 kings are drawn and at least one king is drawn" ≡ "1 king is drawn or 2 kings are drawn" ≡

"1 king is drawn from 4 kings and the other 4 cards are drawn from the remaining 48, or 2 kings are drawn from 4 kings and the other 3 cards are drawn from the remaining 48"

#(A∩B) = number of ways 1 king is drawn from 4 kings times the number of ways the other 4 cards are drawn from the remaining 48, plus the number of ways 2 kings are drawn from 4 kings times the number of ways the other 3 cards are drawn from the remaining 48.

#(A∩B) = C(4,1)*C(48,4) + C(4,2)*C(48,3) = (4)*(194580) + (6)*(17296) = 882096.

So Pr(A|B) = #(A∩B) / #(B) = 882096/886656 ≈ .995 = 99.5%.

(b) Find the probability that 3 kings and 2 Queens are drawn given that at least 3 cards drawn are of the same number or the same letter.

Let event B be the event "at least 3 cards drawn are of the same number or the same letter."

And let A be the event "3 kings and 2 Queens are drawn."

We want to calculate Pr(3 kings and 2 Queens are drawn given that at least 3 cards drawn are of the same number or the same letter) = Pr(A | B) = #(A∩B) / #(B).

First figuring out #(B).

B ≡ "at least 3 cards drawn are of the same number or the same letter" ≡ "3 cards drawn are of the same number or letter or 4 cards drawn are drawn of the same number or letter" ≡ "a letter or number is chosen from 13 different letters or numbers, and 3 cards are drawn of that letter or number and the other 2 cards are drawn from the remaining 48 or 4 cards are drawn of that letter or number and the other 1 card is drawn from the remaining 48"

#(B) = number of ways a letter or number is chosen from 13 different letters or numbers, times number of ways 3 cards are drawn from the 4 cards of that letter or number times number of ways the other 2 cards are drawn from the remaining 48 plus number of ways 4 cards are drawn from the 4 cards of that letter or number times number of ways the 1 other card is drawn from the remaining 48.

To put more succinctly,

#(B) = C(13,1)*(C(4,3)*C(48,2) + C(4,4)*C(48,1)) = 13*((4)*(1128) + (1)*(48)) = 59280.

Now figuring out #(A∩B).

A∩B ≡ "3 kings and 2 Queens are drawn"

#(A∩B) = number of ways 3 kings are drawn from 4 kings times number of ways 2 queens are drawn from 4 queens = C(4,3)*C(4,2) = 4*6 = 24.

Pr(A|B) = #(A∩B) / #(B) = 24/59280 ≈ 4.05×10⁻⁴ ≈ .04%