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Braden bought new equipment for his bowling alley, including a ball return machine. There is a 66%66\% chance that the machine returns a bowling ball with the finger holes facing up. If the machine returns 55 bowling balls, what is the probability that exactly 44 will have the finger holes facing up? Write your answer as a decimal rounded to the nearest thousandth. ____

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Q. Braden bought new equipment for his bowling alley, including a ball return machine. There is a 66%66\% chance that the machine returns a bowling ball with the finger holes facing up. If the machine returns 55 bowling balls, what is the probability that exactly 44 will have the finger holes facing up? Write your answer as a decimal rounded to the nearest thousandth. ____
  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 on a single trial, and (nk)\binom{n}{k} is the binomial coefficient.
  2. Calculate Binomial Coefficient: First, let's calculate the binomial coefficient for 55 choose 44, which is 5!4!(5−4)!=51=5\frac{5!}{4!(5-4)!} = \frac{5}{1} = 5.
  3. Plug in Values: Now, we plug in the values into the binomial formula. The probability of success on a single trial pp is 66%66\% or 0.660.66, and the probability of failure 1−p1-p is 34%34\% or 0.340.34. So, P(4)=(54)×(0.66)4×(0.34)5−4P(4) = \binom{5}{4} \times (0.66)^4 \times (0.34)^{5-4}.
  4. Calculate Probability: Calculating P(4)P(4) gives us P(4)=5×(0.66)4×(0.34)1P(4) = 5 \times (0.66)^4 \times (0.34)^1.
  5. Final Calculation: Now, let's do the math: P(4)=5×(0.66)4×(0.34)=5×0.18974736×0.34=0.32252829664P(4) = 5 \times (0.66)^4 \times (0.34) = 5 \times 0.18974736 \times 0.34 = 0.32252829664.

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