An urn contains 7 red and 5 green balls. Four balls are randomly drawn from the urn in succession, with replacement. That is, after each draw, the selected ball is returned to the urn. What is the probability that all 4 balls drawn from the urn are red? Round your answer to three decimal places.(If necessary, consult a list of formulas.)
Question
An urn contains 7 red and 5 green balls. Four balls are randomly drawn from the urn in succession, with replacement. That is, after each draw, the selected ball is returned to the urn. What is the probability that all 4 balls drawn from the urn are red? Round your answer to three decimal places.(If necessary, consult a list of formulas.)
Solution
To find the probability that all 4 balls drawn from the urn are red, we can use the concept of independent events.
The probability of drawing a red ball on any single draw is given by the ratio of the number of red balls to the total number of balls in the urn. In this case, there are 7 red balls and 12 total balls (7 red + 5 green). Therefore, the probability of drawing a red ball on any single draw is 7/12.
Since the draws are made with replacement, the probability of drawing a red ball on each of the four draws is independent of the previous draws. Therefore, we can multiply the probabilities of each individual draw to find the probability of all four draws being red.
So, the probability of all four balls being red is (7/12) * (7/12) * (7/12) * (7/12) = (7/12)^4.
Calculating this probability, we get (7/12)^4 ≈ 0.091.
Therefore, the probability that all 4 balls drawn from the urn are red is approximately 0.091, rounded to three decimal places.
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