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During a single day at radio station WMZH, the probability that a particular song is played is
1//6. What is the probability that this song will be played on exactly 3 days out of 4 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 $1 / 6$ . What is the probability that this song will be played on exactly $3$ days out of $4$ 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

1 / 6

. What is the probability that this song will be played on exactly

3

days out of

4

days? Round your answer to the nearest thousandth.

\newline

Answer:

Understand and Determine Probability: Understand the problem and determine the probability of the song being played on any given day. $\newline$ The probability of the song being played on a single day is given as $\frac{1}{6}$ . We need to find the probability of this event happening exactly $3$ times in a sequence of $4$ days.
Use Binomial Probability Formula: Use the binomial probability formula to calculate the probability of the song being played exactly $3$ times in $4$ days. $\newline$ The binomial probability formula is $P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$ , where: $\newline$ - $P(X=k)$ is the probability of $k$ successes in $n$ trials, $\newline$ - $\binom{n}{k}$ is the binomial coefficient, $\newline$ - $p$ is the probability of success on a single trial, and $\newline$ - $(1-p)$ is the probability of failure on a single trial. $\newline$ In this case, $n=4$ , $4$ $0$ , and $4$ $1$ .
Calculate Binomial Coefficient: Calculate the binomial coefficient $\binom{4}{3}$ . $\binom{4}{3} = \frac{4!}{3! \cdot (4-3)!} = \frac{4}{1} = 4$ .
Calculate Probability of $3$ Times: Calculate the probability of the song being played exactly $3$ times. $\newline$ Using the binomial probability formula: $\newline$ $P(X=3) = \binom{4}{3} \times \left(\frac{1}{6}\right)^3 \times \left(\frac{5}{6}\right)^{4-3}$ $\newline$ $P(X=3) = 4 \times \left(\frac{1}{6}\right)^3 \times \left(\frac{5}{6}\right)^1$
Perform Calculations: Perform the calculations. $\newline$ $P(X=3) = 4 \times \left(\frac{1}{216}\right) \times \left(\frac{5}{6}\right)$ $\newline$ $P(X=3) = 4 \times \frac{1}{216} \times \frac{5}{6}$ $\newline$ $P(X=3) = \frac{20}{1296}$
Simplify Fraction and Round: Simplify the fraction and round to the nearest thousandth. $\newline$ $\frac{20}{1296}$ can be simplified to $\frac{5}{324}$ . $\newline$ To round to the nearest thousandth, we convert $\frac{5}{324}$ to a decimal. $\newline$ $\frac{5}{324} \approx 0.015432098765432098...$ $\newline$ Rounded to the nearest thousandth, this is approximately $0.015$ .

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