Answer to Question #294059 in Statistics and Probability for arianne

Question

Two cards are drawn from a deck. How many possible values can each of the following variables take?

• Sum of the numbers on the cards:
• Number of times both cards are black:
• Number of times both cards are 7s:
• Number of times the first card is six and the second card is red:
• Number of times the first card is face card and the second card is not a face card:

Answer 1

$a)$

In a deck of 52 cards, there are cards numbered from 1 to 13.

When two cards are drawn, least sum obtained is 2 (=1+1) and greatest sum obtained is 26 (=13+13).

$b)$

There are 26 black cards.

Since two cards are drawn, then we have $\binom{26}{2}={26\times25\over2}=325$

Therefore, there are 325 ways in which both cards are black

$c)$

There are four 7's and the number of ways in which both cards are 7's is $\binom{4}{2}=2\times3=6$ ways

$d)$

There are four 6's and 26 red cards. The number of  the first card is six and the second card is red is,

$\binom{4}{1}\times\binom{26}{1}=4\times26=104$ times

$e)$

There are 12 face cards and 40 non-faced cards.

So, number of times the first card is face card and the second card is not a face card from deck of 52 cards is $\binom{12}{1}\times \binom{40}{1} =12\times40=480$ times.