Probability of Choosing a Face Card

Date: 05/08/2001 at 09:48:40
From: Timi
Subject: Probability of getting a face card 3 times in a row...

Hello! 

Not long ago, I ran into this problem:

"If you pick a card from a regular deck of cards (52), what is the 
probability of getting a face card three times in a row? Assume that 
the face card is put back in after each draw, so that you are 
selecting from a full deck each time."

The way I tried to reason was that there are 16 face cards in a 52 
deck, so the chance of picking a face card is  16/52 = about 30% 
a time. Unfortunately, I was stuck here... Could you show me 
how to solve this problem?

Date: 05/08/2001 at 15:36:08
From: Doctor Twe
Subject: Re: Probability of getting a face card 3 times in a row...

Hi Timi - thanks for writing to Dr. Math.

Usually, only Jacks, Queens, and Kings are considered "face cards," 
so the probability of picking a face card on the first draw would be 
12/52, or 3/13 (about 23%). But if you're counting Aces as faces (as 
they sometimes are), then your 16/52 = 4/13 (30.77%) is correct.

Now, when we have two or more independent events (that is, the outcome 
of one event does not affect the probability of the next event), to 
find the probability of all occurring, we multiply the probabilities 
together. So the probability of getting two face cards in a row 
(counting the Ace as a face card) would be:

     4/13 * 4/13 = 16/169   (approx. 9.5%)

and 3 face cards in a row would be:

     4/13 * 4/13 4/13 = 64/2197   (approx. 2.9%)

Note that if the face cards are *not* reshuffled in the deck, then the 
events are not independent (drawing the first card changes the 
probability for the second draw, because there are different numbers 
of cards and faces left), and we have to perform a different 
calculation to get the probability.

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/   

Date: 05/08/2001 at 17:43:23
From: Timea Agnew
Subject: Re: Probability of getting a face card 3 times in a row...

Thank you VERY much for your help!

Timi