barcode font reporting services Basic Probability in Software

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CHAPTER 1 Basic Probability
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we shall assume that P(A wins any one game) (a) P(A wins all 3 games) 6 12 1 , 2 P(B wins any one game) P(A1) P(A2) P(A3) 1 1 1 2 2 2 4 12 1 3 1 8
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P(A1 > A2 > A3)
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assuming that the results of each game are independent of the results of any others. (This assumption would not be justifiable if either player were psychologically influenced by the other one s winning or losing.)
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1 5 (b) In any one game the probability of a nondraw (i.e., either A or B wins) is q 1 2 3 6 and the 1 probability of a draw is p 1 q 6. Then the probability of 2 draws in 3 trials is (see Problem 1.37)
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P(A and B win alternately)
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3 p2 q3 2
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P(A wins then B wins then A wins
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or B wins then A wins then B wins)
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P(A1 > B2 > A3) P(A1)P(B2)P(A3) 1 1 1 2 3 2 1 1 1 1 P(B1 > A2 > B3) 1 1 1 3 2 3 P(B1)P(A2)P(B3) 5 36
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1 5 3 6 6
5 72
P(B wins at least one game)
P(B wins no game) P(Br > Br > Br ) 1 2 3 2 2 2 3 3 3
P(Br ) P(Br ) P(Br ) 1 2 3 19 27
1.40. A and B play a game in which they alternately toss a pair of dice. The one who is first to get a total of 7 wins the game. Find the probability that (a) the one who tosses first will win the game, (b) the one who tosses second will win the game.
(a) The probability of getting a 7 on a single toss of a pair of dice, assumed fair, is 1 > 6 as seen from Problem 1.9 and Fig. 1-9. If we suppose that A is the first to toss, then A will win in any of the following mutually exclusive cases with indicated associated probabilities: 1 (1) A wins on 1st toss. Probability . 6 5 5 1 (2) A loses on 1st toss, B then loses, A then wins. Probability . 6 6 6 (3) A loses on 1st toss, B loses, A loses, B loses, A wins. Probability 1 6 5 5 1 6 6 6 5 5 5 5 1 6 6 6 6 6 5 6
5 5 5 5 1 . 6 6 6 6 6 ................................................................................ Then the probability that A wins is c 1 B1 6 5 6 c R 5 2 a b 6 5 11
1>6 (5>6)2
6 11
where we have used the result 6 of Appendix A with x (b) The probability that B wins the game is similarly 5 1 a ba b 6 6 5 5 5 1 a ba ba ba b 6 6 6 6 c
(5 > 6)2.
5 1 a b a b c1 6 6 5>36 1 (5>6)2
5 4 a b 6
CHAPTER 1 Basic Probability
Therefore, we would give 6 to 5 odds that the first one to toss will win. Note that since 6 5 1 11 11 the probability of a tie is zero. This would not be true if the game was limited. See Problem 1.100.
1.41. A machine produces a total of 12,000 bolts a day, which are on the average 3% defective. Find the probability that out of 600 bolts chosen at random, 12 will be defective.
Of the 12,000 bolts, 3%, or 360, are defective and 11,640 are not. Then: Required probability
360C12 11,640C588 12,000C600
1.42. A box contains 5 red and 4 white marbles. Two marbles are drawn successively from the box without replacement, and it is noted that the second one is white. What is the probability that the first is also white Method 1
If W1, W2 are the events white on 1st draw, white on 2nd draw, respectivley, we are looking for P(W1 u W2). This is given by P(W1 > W2) (4>9)(3>8) 3 P(W1 u W2) P(W2) 8 4>9
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