Marbles and DiceDate: 08/11/97 at 17:57:50 From: Connie Subject: Marbles and dice 1. A jar has three black marbles and five red marbles. What is the probability that one red and one black marble will be picked without replacing the marbles? 2. Two ordinary dice are thrown. What is the probability that the sum will be six or at least one of the two faces will be odd? Thanks, Connie Date: 08/12/97 at 07:06:18 From: Doctor Anthony Subject: Re: Marbles and dice Use the nCr notation, where this symbol stands for the number of ways of selecting a group of r things from n different things. For example 7C3 is given by: 7! 7 x 6 x 5 7C3 = ------ = 35 It can also be calculated from ----------- 3! 4! 1 x 2 x 3 In this problem we must select 1 red from 5 red and 1 black from 3 black. This is 5C1 x 3C1 = 5 x 3 = 15 However, the number of ways of selecting any 2 marbles from 8 is given by 8C2 = (8 x 7)/(1 x 2) = 28 So the probability of 1 red, 1 black = 15/28 An alternative method for this relatively simple problem is to think that we could draw red, then black with probability 5/8 x 3/7 = 15/56 or, alternatively, we could draw black then red with probability 3/8 x 5/7, which again gives 15/56. Total probability = 30/56 = 15/28 as before. >2. Two ordinary dice are thrown. What is the probability that the sum >will be six or at least one of the two faces will be odd? Scores Probability -------- ---------------- You get sum = 6 in the following ways 5 + 1 1/6 x 1/6 = 1/36 4 + 2 1/36 3 + 3 1/36 2 + 4 1/36 1 + 5 1/36 So the probability that sum = 6 is 5/36 Note that in three of these cases at least one face is odd, so the probability that sum = 6 AND at least one face is odd is 3/36 The probability, in general, that if we throw two dice at least one face will be odd can be calculated from odd, odd = 1/2 x 1/2 = 1/4 odd, even = 1/4 even, odd = 1/4 Giving a total of 3/4. Alternatively, it is 1 - probability both even = 1 - 1/4 = 3/4 Finally we use the addition rule in the form P(A or B) = P(A) + P(B) - P(A and B) P(sum = 6 or at least 1 face odd) = 5/36 + 3/4 - 3/36 = 29/36 -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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