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It is believed that $71\%$ of tickets purchased for an upcoming music festival are actually legitimate tickets. The rest are fake tickets sold by scam artists. Assume everyone who bought a ticket shows up at the concert. $\newline$ If security guards at the festival randomly select $5$ tickets to examine in more detail, what is the probability that exactly $4$ of the tickets are legitimate? $\newline$ Write your answer as a decimal rounded to the nearest thousandth. $\newline$ ____ $\newline$

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Find probabilities using the binomial distribution

Full solution

Q. It is believed that

71\%

of tickets purchased for an upcoming music festival are actually legitimate tickets. The rest are fake tickets sold by scam artists. Assume everyone who bought a ticket shows up at the concert.

\newline

If security guards at the festival randomly select

5

tickets to examine in more detail, what is the probability that exactly

4

of the tickets are legitimate?

\newline

Write your answer as a decimal rounded to the nearest thousandth.

\newline

____

\newline

Use Binomial Probability Formula: We need to use the binomial probability formula, which is $P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$ , where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success, and $\binom{n}{k}$ is the binomial coefficient.
Calculate Binomial Coefficient: First, calculate the binomial coefficient for $5$ choose $4$ . This is $\frac{5!}{4! \times (5-4)!}$ , which simplifies to $5$ .
Determine Probabilities: The probability of success (legitimate ticket) is $0.71$ , and the probability of failure (fake ticket) is $1 - 0.71 = 0.29$ .
Apply Binomial Formula: Now plug the values into the binomial formula: $P(4 \text{ legit tickets}) = \binom{5}{4} \times (0.71^4) \times (0.29^1)$ .
Calculate Probability: Calculate the probability: $P(4 \text{ legit tickets}) = 5 \times (0.71^4) \times (0.29)$ .
Perform Calculations: Perform the calculations: $P(4 \text{ legit tickets}) = 5 \times (0.25411681) \times (0.29)$ .
Finish Calculation: Finish the calculation: $P(4 \text{ legit tickets}) = 5 \times 0.0737138779$ .
Multiply by $5$ : Finally, multiply by $5$ : $P(4 \text{ legit tickets}) = 0.3685693895$ .
Round to Nearest Thousandth: Round the answer to the nearest thousandth: $P(4 \text{ legit tickets}) = 0.369$ .

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