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During a single day at radio station WMZH, the probability that a particular song is played is 2//3. What is the probability that this song will be played on exactly 2 days out of 5 days? Round your answer to the nearest thousandth. Answer:

During a single day at radio station WMZH, the probability that a particular song is played is 2/3 2 / 3 . What is the probability that this song will be played on exactly 22 days out of 55 days? Round your answer to the nearest thousandth.\newlineAnswer:

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Q. During a single day at radio station WMZH, the probability that a particular song is played is 2/3 2 / 3 . What is the probability that this song will be played on exactly 22 days out of 55 days? Round your answer to the nearest thousandth.\newlineAnswer:
  1. Identify Problem Type: Identify the type of probability problem. We are dealing with a binomial probability problem where there are a fixed number of trials (55 days), two possible outcomes (the song is played or not played), and the probability of success (the song being played) is constant (23\frac{2}{3}).
  2. Use Binomial Probability Formula: Use the binomial probability formula.\newlineThe binomial probability formula is P(X=k)=(nk)⋅pk⋅(1−p)n−kP(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}, where P(X=k)P(X=k) is the probability of kk successes in nn trials, pp is the probability of success on a single trial, and (nk)\binom{n}{k} is the binomial coefficient.
  3. Calculate Binomial Coefficient: Calculate the binomial coefficient (52)\binom{5}{2}.(52)=5!2!⋅(5−2)!=1202⋅6=12012=10\binom{5}{2} = \frac{5!}{2! \cdot (5-2)!} = \frac{120}{2 \cdot 6} = \frac{120}{12} = 10.
  4. Calculate Probability of 22 Days: Calculate the probability of the song being played exactly 22 days out of 55. Using the binomial probability formula, P(X=2)=(52)×(23)2×(13)5−2P(X=2) = \binom{5}{2} \times \left(\frac{2}{3}\right)^2 \times \left(\frac{1}{3}\right)^{5-2}.
  5. Perform Calculations: Perform the calculations.\newlineP(X=2)=10×(23)2×(13)3=10×(49)×(127)=10×(4243)=40243P(X=2) = 10 \times \left(\frac{2}{3}\right)^2 \times \left(\frac{1}{3}\right)^3 = 10 \times \left(\frac{4}{9}\right) \times \left(\frac{1}{27}\right) = 10 \times \left(\frac{4}{243}\right) = \frac{40}{243}.
  6. Round to Nearest Thousandth: Round the answer to the nearest thousandth.\newlineP(X=2)≈40243≈0.164609053≈0.165P(X=2) \approx \frac{40}{243} \approx 0.164609053 \approx 0.165 when rounded to the nearest thousandth.

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