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A psychology professor surveyed her class and determined that $59\%$ of the students have acrophobia, a fear of heights. If $2$ of the students are chosen at random, what is the probability that $0$ have acrophobia? $\newline$ Write your answer as a decimal rounded to the nearest thousandth.____

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Find probabilities using the binomial distribution

Full solution

Q. A psychology professor surveyed her class and determined that

59\%

of the students have acrophobia, a fear of heights. If

2

of the students are chosen at random, what is the probability that

0

have acrophobia?

\newline

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

Use Binomial Probability Formula: Use the binomial probability formula: $P(X = k) = C(n, k) \cdot (p)^k \cdot (1-p)^{(n-k)}$ . Here, $n = 2$ , $k = 0$ , and $p = 0.59$ .
Calculate Combination: Calculate $C(2, 0)$ which is $\frac{2!}{0! (2 - 0)!}$ . $C(2, 0) = 1$ .
Solve $(0.59)^0$ : Solve $(0.59)^0$ which is $1$ because any number to the power of $0$ is $1$ .
Simplify $(1 - 0.59)^{(2 - 0)}$ : Simplify $(1 - 0.59)^{(2 - 0)}$ which is $(0.41)^2$ .
Calculate $(0.41)^2$ : Calculate $(0.41)^2$ . $(0.41)^2 = 0.1681$ .
Multiply Values Together: Multiply all the values together: $P(X = 0) = 1 \times 1 \times 0.1681$ . $\newline$ $P(X = 0) = 0.1681$ .
Round to Nearest Thousandth: Round the answer to the nearest thousandth. $\newline$ $P(X = 0) = 0.168$ .

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