Answer to Question #302157 in Statistics and Probability for kazuha

Question

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

1. sum of the numbers on the cards ___________

2. number of times both cards are black ________

3. Number of times both cards are 7s ___________

4. Number of times the first card is six and the second card is red _____

5. Number of times the first card is face card and the second card is not

a face card _______

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Answer 1

1.

In a deck of 52 cards, there are 40 cards numbered from 2 to 10.

When we draw two cards, least sum obtained is 4 (=2+2 and greatest sum obtained is 20 (=10+10).

There are 17 possible values of Sum of the numbers on the cards:

b) There are 13 black cards. First card will be black in 13 cases. If the first card is black, then the second card will be black in $13-1=12$ cases. Since the events are independent then both cards are black in $13(12)=156$ cases.

156 times both cards are black.

c)There are four 7s. First card will be 7s in 4 cases. If the first card is 7s, then the second card will be 7s in $4-1=3$ cases. Since the events are independent then both cards are 7s in $4(3)=12$ cases.

12 times both cards are 7s.

d) There are four 6s. There are 13 red cards.

First card will be 6s in 4 cases. It may be red 6s in 2 cases or black 6s in 2 cases.

If the first card is red 6s, then the second card will be red in $13-1=12$ cases. Since the events are independent then the first card is red 6s and the second card is red in $2(12)=24$ cases.

If the first card is black 6s, then the second card will be red in $13$ cases. Since the events are independent then the first card is black 6s and the second card is red in $2(13)=26$ cases.

Hence the first card is 6s and the second card is red in $24+26=50$ cases.

50 times the first card is six and the second card is red.

e) First card will be face card in 12 cases. The second card will be not face card in 40 cases. Since the events are independent then the first card is face card and the second card is not a face card in $12(40)=480$ cases.

480 times the first card is face card and the second card is not a face card.