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During a single day at radio station WMZH, the probability that a particular song is played is 5/8. What is the probability that this song will be played on exactly 6 days out of 7 days? Round your answer to the nearest thousandth.

User ArdaZeytin
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1 Answer

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Final answer:

To find the probability of a song being played on exactly 6 days out of 7 days at a radio station, we use the binomial probability formula with a success probability of 5/8 each day. The calculated probability, after applying the formula and rounding to the nearest thousandth, is approximately 0.609.

Step-by-step explanation:

The question deals with the probability that a particular song will be played on exactly 6 days out of 7 days. The probability that the song will be played in any given day is given as 5/8 and the complement, the probability that the song will not be played on any given day, is 1 - 5/8 = 3/8. The situation can be described by a binomial distribution where there are 7 trials (days), we want exactly 6 successes (days the song is played), and the probability of success on each trial is 5/8.

To find the probability of exactly 6 successes, we use the binomial probability formula:

P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)

Where:

  • C(n, k) is the combination of n things taken k at a time
  • p is the probability of success on a single trial
  • k is the number of successes (which is 6 for this problem)
  • n is the number of trials (which is 7 for this problem)

Calculating this gives:

P(X = 6) = C(7, 6) * (5/8)^6 * (3/8)^(7-6)

P(X = 6) = 7 * (5/8)^6 * (3/8)^1

P(X = 6) = 7 * 0.232 * 0.375

P(X = 6) is approximately 0.609 (rounded to the nearest thousandth).

User Fatemeh Rostami
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