Two cards are drawn from a standard deck of cards. Find each probability: (See in the Description)

Question

1. P(both kings or both face cards) ans. 11/221

 

2. P(both either red or a king) ans. 63/221

 

I have the correct answers for these events in Adding Probabilities, but I am a substitute teacher, my degree is in music. I need to help the students to use the correct process to arrive at the answers.

Thank you in advance for any help you can give me.

Answer 1

My sympathies!!

There's a whole concept here of events, and sets of events, and subsets, etc., which would take more time to go through.  I'll try to cut to the chase without any serious "hand waving."

 

1. First of all, notice that "both kings" is just a subset of "both face cards."  That is, if both are kings, then they are also both face cards.  So you really only have to work out the probability of both being face cards, and the kings will come along with that.

 

So first of all, let's get the basic numbers:  There are 52 cards in the deck, and 12 of these are face cards -- J, Q, K, all 4 suits.  So the probability of drawing a face card in the first draw is 12/52 = 3/13.

 

After the first card is drawn, however, there are only 51 cards left.  And if you're still in line to have both be face cards, then the first one had to be a face card, which means there are only 11 of those left.  So if you're going ahead with the second draw, the probability of getting a face card is now 11 / 51.

 

To calculate the probability that BOTH are face cards, you multiply the two probabilities together:  3/13 * 11/51.  Multiply those two fractions together, and cancel out the common factors, and you get 11/221

 

2. How many cards would make the conditions true (red or king)?  Well, there are 26 red cards, and there are 4 kings.  So that's 30.  BUT, there are two cards that are both red AND king.  So if you just add the number of red cards to the number of kings, you are "double counting" -- counting these cards in both categories.  So you have to subtract out the 2 cards that are both.  That gives you 28.

 

So the probability of getting a red or king on the first draw is 28/52 = 7/13.

 

If you succeeded on the first draw and you are continuing on to the second, you have to do what we did in the first problem -- account for the card that has already been drawn.  So there will be 51 cards left, of which only 27 will meet the required conditions.  So the probability that this one will be red or king is 27/51 = 9/17. 

 

So again, multiply 7/13 by 9/17 and you get 63/221.