An urn contains 8 white balls and 10 red balls. If Frank chooses 8 balls at random from the urn, what is the probability that he will select 5 white balls and 3 red balls? Round your answer to 3 decimal places.(If necessary, consult a list of formulas.)
Question
An urn contains 8 white balls and 10 red balls. If Frank chooses 8 balls at random from the urn, what is the probability that he will select 5 white balls and 3 red balls? Round your answer to 3 decimal places.(If necessary, consult a list of formulas.)
Solution
To solve this problem, we need to use the combination formula which is C(n, r) = n! / [r!(n-r)!], where n is the total number of items, r is the number of items to choose, and "!" denotes factorial.
Step 1: Calculate the total number of ways to draw 8 balls out of 18 (8 white + 10 red). This is C(18, 8).
Step 2: Calculate the number of ways to draw 5 white balls out of 8. This is C(8, 5).
Step 3: Calculate the number of ways to draw 3 red balls out of 10. This is C(10, 3).
Step 4: Multiply the results from steps 2 and 3 together. This gives the number of ways to draw 5 white balls and 3 red balls.
Step 5: Divide the result from step 4 by the result from step 1. This gives the probability of drawing 5 white balls and 3 red balls.
Let's calculate:
Step 1: C(18, 8) = 18! / [8!(18-8)!] = 43758
Step 2: C(8, 5) = 8! / [5!(8-5)!] = 56
Step 3: C(10, 3) = 10! / [3!(10-3)!] = 120
Step 4: 56 * 120 = 6720
Step 5: 6720 / 43758 = 0.1535
So, the probability that Frank will select 5 white balls and 3 red balls is approximately 0.154 when rounded to three decimal places.
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