probability of deck of cards

Question

A standard deck of cards has had all the face cards (jacks, queens, and

kings) removed so that only the ace through ten of each suit remains. A

game is played in which two cards are drawn (without replacement) from

this deck and a six-sided die is rolled. For the purpose of this game, an ace

is considered to have a value of 1.

(a) Determine the total number of possible outcomes for this game.

(b) Find the probability of each of the following events.

(i) one even card is drawn and an even number is rolled

(ii) two even cards are drawn and an odd number is rolled

(iii) one card of 3 is drawn and 3 or less is rolled

(iv) the sum of the cards and the die is 7

(v) the sum of the cards and the die is less than 5

Answer 1

In none of these problems is the order in which the card is drawn important, so we look at the number of combinations.

a)The straightforward answer, as there are 40 cards left, is C(40,2)*6 = 780 *6 = 4680

b)

1) There are 20 odds and 20 evens, and 3 odd and 3 even rolls of the die.

To draw one even card and one even roll, there are C(20,1)C(20,1)*3 =1200 ways

1200/4680 = 10/39

2) 2 even and an odd C(20,2)*3 = 190*3=570

570/4680 = 19/156

3) One card of 3 is drawn and one roll of 3 or less

There are 4 3s and 40-4= 36 non-3s,and 3 rolls of 3 or less

C(4,1)C(36,1)*3 = 4*36*3 = 432

432/4680 = 6/65

4) The sum of the cards and die is 7

We have the same number of ways when the 2 cards are different (there is one roll that then adds to 7) and the same number of ways when the 2 cards are the same (once again, there is one roll that adds to 7.

If 2 cards are different, there are C(4,1)*C(4,1) = 16 ways. If 2 cards are the same, there are C(4,2) = 6 ways. Note that the sum of the two cards must be less than 7, as the die is at least 1.

Different:(1,2), (1,3), (1,4),(1,5),(2,3),(2,4) Rolls are then 4, 3, 2, 1, 2, 1

The same: (1,1), (2,2), (3,3) - the roll is then 5, 3, and 1

6*16+3*6 = 114

114/4680 = 19/780

5) Less than 5 means 3 or 4

There can be 2 aces and a roll of 1 or 2, and an ace, a 2, and a roll of 1.

C(4,2)*2 + C(4,1)C(4,1) =12 +16 =28

28/4680 = 7/1170