$$ \begin{array}{l} ext { (e) } n=5, p=0.68, X=2 \ P(X)=\square\end{array} $$

Question

$$ \begin{array}{l} 	ext { (e) } n=5, p=0.68, X=2 \ P(X)=\square\end{array} $$

UpStudy AI · Accepted Answer

We are given a binomial probability problem with
$$
n = 5,\quad p = 0.68,\quad \text{and } X = 2.
$$

The probability of exactly $$X=2$$ successes in $$n$$ independent trials is given by the binomial probability formula:
$$
P(X=2)=\binom{5}{2} (0.68)^2 (1-0.68)^{5-2}.
$$

Step 1: Compute the binomial coefficient
$$
\binom{5}{2}=\frac{5!}{2!(5-2)!}=\frac{5 \times 4}{2 \times 1}=10.
$$

Step 2: Calculate the probability for 2 successes
$$
(0.68)^2 = 0.68 \times 0.68 = 0.4624.
$$

Step 3: Calculate the probability for 3 failures
$$
1-0.68=0.32,\quad \text{and}\quad (0.32)^3 = 0.32 \times 0.32 \times 0.32 \approx 0.032768.
$$

Step 4: Combine the results
$$
P(X=2)=10 \times 0.4624 \times 0.032768 \approx 10 \times 0.0151518 = 0.151518.
$$

Thus, the probability is
$$
P(X=2) \approx 0.1515.
$$
The probability $$ P(X=2) $$ is approximately 0.1515.

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