How do I calculate probability of face cards with a tree diagram?

Question

In poker we are dealt two cards from the deck (without replacement). I am interested in the number of ’face cards’ I get. We define a J, Q, K and Ace to be ’face cards’. Construct a tree diagram for the experiment showing all final outcomes with attached probabilitites. Use this to find the probability that exactly one of our two cards is a face card.

Answer 1

Assuming the Ace is a "face card", which is debatable by definition, however . . . ):

Let "F" mean a Face Card is drawn and let "N" mean No Face Card is drawn:

Having exactly one face card can occur as FN or as NF so 48/221 + 48/221 = 96/221

Answer 2

Hi Chad,

There are two possible outcomes per draw: face and non-face. In a 52-card deck, there are 4 face cards per suit for a total of 16 cards. So the probability of a face card on the first draw is 16/52 or 4/13. So, the first branches of your tree diagram would be 16/52 (for face) and 36/52 (for non). Now, the second branch depends on the outcome of the first draw. It is difficult to draw a tree diagram here in Wyzant, so I will explain without.

There are two ways you can get exactly one face card: drawing a face on the first draw and a non-face on the second and drawing a non-face on the first draw and a face on the second. Now, we already know the probability of a face on the first draw is 16/52. If we want only one face, this scenario requires a non-face on the second draw. This probability would be 36/51. We removed a card and are assuming it was a face. So, using the Multiplication Rule:

P1= (16/52)*(36/51) = 0.217

Now, we need to address the other scenario—non-face on draw 1 and face on draw 2. So, we already know probability of non-face on draw 1 is 36/52. The probability of a face on the second draw, then, would be 16/51. We removed a card and are assuming it is a non-face. Again, use the Multiplication Rule:

P2= (36/52)*(16/51) = 0.217

Add these probabilities together:

P = 0.217 + 0.217

P(Exactly one face) = 0.434

I hope this helps.