CHAPTER 1 Basic Probability

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Using subscripts 1, 2 to denote 1st and 2nd tosses of the dice, we see that the probability of no 7 or 11 on either the first or second tosses is given by P(Ar ) P(Ar u Ar ) 1 2 1 using the fact that the tosses are independent. P(Ar ) P(Ar2 ) 1 7 7 9 9 49 , 81

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1.10. Two cards are drawn from a well-shuffled ordinary deck of 52 cards. Find the probability that they are both aces if the first card is (a) replaced, (b) not replaced. Method 1

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Let A1 event ace on first draw and A2 P(A1) P(A2 u A1). event ace on second draw. Then we are looking for P(A1 > A2)

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(a) Since for the first drawing there are 4 aces in 52 cards, P(A1) 4 > 52. Also, if the card is replaced for the second drawing, then P(A2 u A1) 4 > 52, since there are also 4 aces out of 52 cards for the second drawing. Then P(A1 > A2) P(A1) P(A2 u A1) 4 4 52 52 1 169

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(b) As in part (a), P(A1) 4 > 52. However, if an ace occurs on the first drawing, there will be only 3 aces left in the remaining 51 cards, so that P(A2 u A1) 3 > 51. Then P(A1 > A2) P(A1) P(A2 Z A1) 4 3 52 51 1 221

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Method 2

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(a) The first card can be drawn in any one of 52 ways, and since there is replacement, the second card can also be drawn in any one of 52 ways. Then both cards can be drawn in (52)(52) ways, all equally likely. In such a case there are 4 ways of choosing an ace on the first draw and 4 ways of choosing an ace on the second draw so that the number of ways of choosing aces on the first and second draws is (4)(4). Then the required probability is (4)(4) (52)(52) 1 169

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(b) The first card can be drawn in any one of 52 ways, and since there is no replacement, the second card can be drawn in any one of 51 ways. Then both cards can be drawn in (52)(51) ways, all equally likely. In such a case there are 4 ways of choosing an ace on the first draw and 3 ways of choosing an ace on the second draw so that the number of ways of choosing aces on the first and second draws is (4)(3). Then the required probability is (4)(3) (52)(51) 1 221

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1.11. Three balls are drawn successively from the box of Problem 1.7. Find the probability that they are drawn in the order red, white, and blue if each ball is (a) replaced, (b) not replaced.

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Let R1 event red on first draw, W2 require P(R1 > W2 > B3). event white on second draw, B3 event blue on third draw. We

(a) If each ball is replaced, then the events are independent and P(R1 > W2 > B3) P(R1) P(W2 u R1) P(B3 u R2 > W2) P(R1) P(W2) P(B3) 6 6 4 5 6 4 4 5

8 225

CHAPTER 1 Basic Probability

(b) If each ball is not replaced, then the events are dependent and P(R1 > W2 > B3) P(R1) P(W2 u R1) P(B3 u R1 > W2) 6 6 4 5 5 4 4 5 5 5 3 5

4 91

1.12. Find the probability of a 4 turning up at least once in two tosses of a fair die.

Let A1 event 4 on first toss and A2 A1 < A2 event 4 on second toss. Then

event 4 on first toss or 4 on second toss or both event at least one 4 turns up,

and we require P(A1 < A2).

Method 1

Events A1 and A2 are not mutually exclusive, but they are independent. Hence, by (10) and (21), P(A1 < A2) P(A1) P(A1) 1 6 1 6 P(A2) P(A2) P(A1 > A2) P(A1) P(A2) 11 36 1