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A surfboard factory just discovered a manufacturing error that causes some of the surfboards to crack easily. As a result, the factory recalled all of the surfboards sold in the last year. Each surfboard has a $75\%$ chance of having the defect. $\newline$ If the factory recalled $3$ surfboards, what is the probability that $0$ of the boards have the defect? $\newline$ 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. A surfboard factory just discovered a manufacturing error that causes some of the surfboards to crack easily. As a result, the factory recalled all of the surfboards sold in the last year. Each surfboard has a

75\%

chance of having the defect.

\newline

If the factory recalled

3

surfboards, what is the probability that

0

of the boards have the defect?

\newline

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

\newline

____

\newline

Use Binomial Probability Formula: Use the binomial probability formula $P(X = k) = C(n, k) \cdot (p)^k \cdot (1-p)^{(n-k)}$ . Here, $n = 3$ , $k = 0$ , and $p = 0.75$ .
Calculate $C(3, 0)$ : Calculate $C(3, 0)$ which is $\frac{3!}{0! \times (3 - 0)!}$ . This simplifies to $1$ because any number factorial divided by itself is $1$ .
Compute $(0.75)^0$ : Compute $(0.75)^0$ which is $1$ , because any number to the power of $0$ is $1$ .
Calculate $(1 - 0.75)^{(3 - 0)}$ : Calculate $(1 - 0.75)^{(3 - 0)}$ which is $(0.25)^3$ . This is $0.25 \times 0.25 \times 0.25$ .
Multiply Values Together: Multiply all the values together. $P(X = 0) = 1 \times 1 \times (0.25)^3$ .
Solve $(0.25)^3$ : Solve $(0.25)^3$ which is $0.015625$ .
Multiply Values from Step $5$ : Multiply the values from Step $5$ . $P(X = 0) = 1 \times 1 \times 0.015625$ .
Final Probability: The final probability is $0.015625$ . Round this to the nearest thousandth.

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