Question 529924: Suppose that 12 green balls and 13 purple balls are placed in an urn. Two balls are then drawn in succession. What is the probability that both balls drawn have the same color if the first ball is replaced before the second is drawn?
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! For the first draw, there is a total of 25 balls in the urn, 12 green plus 13 purple.
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You have a 12 in 25 chance of drawing a green ball. Dividing 12 by 25 gives you the probability of drawing a green ball as:
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12/25 = 0.48 or 48%
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You then put that green ball back into the urn. On the second draw you again have a 12 in 25 chance of drawing a green ball. So the probability of getting a green ball on this second draw is 0.48 or 48%
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The probability of drawing a green ball for each of the two draws in the product of these two probabilities. So the probability of drawing two green balls on the first two draws from the urn is:
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0.48 times 0.48 = 0.2304
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The problem asks you for the probability of drawing two balls of the same color on two draws. You have calculated the probability of drawing two green balls on the first two draws. But there is a second possibility. You could draw two purple balls in succession. You can calculate that possibility by doing a similar analysis.
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Since 13 of the 25 balls in the urn are purple, the probability of drawing a purple ball on the first draw is:
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13/25 = 0.52 or 52%
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After you throw this purple ball back into the urn, you again have a 13/25 or 0.52 chance of getting a purple ball on the second drawing. So the probability of getting two purple balls on the two draws is the product of these two probabilities or:
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0.52 times 0.52 = 0.2704
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The total probability of drawing two balls of the same color on two draws (with replacement of the first ball drawn) is the sum of the probability of drawing two green balls plus the probability of drawing two purple balls or:
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0.2304 + 0.2704 = 0.5008 = 50.08%
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Hope this helps you to understand the process for solving this problem.
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Just for fun, what is the probability of drawing a green ball on the first draw and a purple ball on the second draw? It is:
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0.48 times 0.52 = 0.2496
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And what is the probability of drawing a purple ball on the first draw and a green ball on the second draw? It is:
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0.52 times 0.48 = 0.2496
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The sum of these two probabilities is the total probability of getting two different color balls on the two draws. Adding these two together results in:
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0.2496 + 0.2496 = 0.4992
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The total probability of either getting two balls of the same color (0.5008) or getting two balls of different color (0.4992) is calculated by adding these two probabilities. When you add them you get a total of 1.0000 or 100%. This check tells you that on two draws you have a 100% chance getting one of the four results: two green; or two purple, or a green then a purple, or a purple then a green.
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