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).