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It is believed that 71%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.\newlineIf security guards at the festival randomly select 55 tickets to examine in more detail, what is the probability that exactly 44 of the tickets are legitimate?\newlineWrite your answer as a decimal rounded to the nearest thousandth.\newline____\newline

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Q. It is believed that 71%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.\newlineIf security guards at the festival randomly select 55 tickets to examine in more detail, what is the probability that exactly 44 of the tickets are legitimate?\newlineWrite your answer as a decimal rounded to the nearest thousandth.\newline____\newline
  1. Use Binomial Probability Formula: We need to use the binomial probability formula, which is P(X=k)=(nk)⋅pk⋅(1−p)n−kP(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}, where nn is the number of trials, kk is the number of successes, pp is the probability of success, and (nk)\binom{n}{k} is the binomial coefficient.
  2. Calculate Binomial Coefficient: First, calculate the binomial coefficient for 55 choose 44. This is 5!4!×(5−4)!\frac{5!}{4! \times (5-4)!}, which simplifies to 55.
  3. Determine Probabilities: The probability of success (legitimate ticket) is 0.710.71, and the probability of failure (fake ticket) is 1−0.71=0.291 - 0.71 = 0.29.
  4. Apply Binomial Formula: Now plug the values into the binomial formula: P(4 legit tickets)=(54)×(0.714)×(0.291)P(4 \text{ legit tickets}) = \binom{5}{4} \times (0.71^4) \times (0.29^1).
  5. Calculate Probability: Calculate the probability: P(4 legit tickets)=5×(0.714)×(0.29)P(4 \text{ legit tickets}) = 5 \times (0.71^4) \times (0.29).
  6. Perform Calculations: Perform the calculations: P(4 legit tickets)=5×(0.25411681)×(0.29)P(4 \text{ legit tickets}) = 5 \times (0.25411681) \times (0.29).
  7. Finish Calculation: Finish the calculation: P(4 legit tickets)=5×0.0737138779P(4 \text{ legit tickets}) = 5 \times 0.0737138779.
  8. Multiply by 55: Finally, multiply by 55: P(4 legit tickets)=0.3685693895P(4 \text{ legit tickets}) = 0.3685693895.
  9. Round to Nearest Thousandth: Round the answer to the nearest thousandth: P(4 legit tickets)=0.369P(4 \text{ legit tickets}) = 0.369.

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