Question

3. Three cards are dealt without replacement, from a well shuffled deck. In answering the following, round your answer to the nearest hundredth of a percent. (a) Find the chance that all of the cards are aces. (b) Find the chance that none of the cards are aces. (c) Find the chance that the cards are not all aces.

Answer 1

There is a total of 52 cards in a regular deck

total of 3 cards are to be chosen from 52, this can be done in  $_{3}^{52}\textrm{C}=22100$ ways

a) There is a total of 4 Aces in a deck

tota number of ways to select 3 cards from 4 is $_{3}^{4}\textrm{C}=4$

the chance that all of the cards are aces=$\frac{4}{22100}=0.00018$

in percentage, the answer is 0.02%

b) the total number of ways to select 3 cards from 48 non-ace cards is $_{3}^{48}\textrm{C}=17296$

The chance that none of the cards are aces = $\frac{17296}{22100}=0.7826$

In percentage, the answer is 78.26%

c) The chance that the cards are not all aces = 1- the chance that all cards are aces = (100 - 0.02) = 99.98%