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During a single day at radio station WMZH, the probability that a particular song is played is 0.68 . 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 $0$ . $68$ . 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. $\newline$ Answer:

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Find probabilities using the binomial distribution

Full solution

Q. During a single day at radio station WMZH, the probability that a particular song is played is

0

.

68

. 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.

\newline

Answer:

Identify Given Probability: Identify the given probability and the event we are interested in. $\newline$ The probability that the song is played on any given day is $0.68$ . We want to find the probability that the song is played exactly on $2$ out of $5$ days.
Recognize Binomial Problem: Recognize that this is a binomial probability problem. We can use the binomial probability formula, which is $P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$ , where: - $P(X=k)$ is the probability of $k$ successes in $n$ trials, - $\binom{n}{k}$ is the binomial coefficient, - $p$ is the probability of success on a single trial, and - $(1-p)$ is the probability of failure on a single trial. In this case, $n=5$ , $k=2$ , and $p=0.68$ .
Calculate Binomial Coefficient: Calculate the binomial coefficient $\binom{5}{2}$ . $\binom{5}{2} = \frac{5!}{2! \cdot (5-2)!} = \frac{120}{2 \cdot 6} = \frac{120}{12} = 10$ .
Calculate Probability of $2$ Days: Calculate the probability of the song being played exactly $2$ days out of $5$ . $\newline$ Using the binomial probability formula: $\newline$ $P(X=2) = \binom{5}{2} \times (0.68)^2 \times (1-0.68)^{5-2}$ $\newline$ $P(X=2) = 10 \times (0.68)^2 \times (0.32)^3$
Perform Calculations: Perform the calculations. $\newline$ $P(X=2) = 10 \times (0.68)^2 \times (0.32)^3$ $\newline$ $P(X=2) = 10 \times 0.4624 \times 0.032768$ $\newline$ $P(X=2) = 10 \times 0.015151104$ $\newline$ $P(X=2) = 0.15151104$
Round to Nearest Thousandth: Round the answer to the nearest thousandth. $\newline$ $P(X=2) \approx 0.152$

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