Question

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Explanation:

Identify the parameters and target

Using the Binomial Probability Distribution knowledge point, we identify the parameters of the binomial random variable \(x\):

• Number of trials: \(n = 9\)
• Probability of success: \(p = 0.4\)
• Probability of failure: \(q = 1 - p = 0.6\)

We need to evaluate the probability:

$$P(2 \le x \le 4)$$

Express the interval probability

Using the Cumulative Binomial Probability knowledge point, we express the probability of the interval as the sum of individual probability mass functions:

$$P(2 \le x \le 4) = P(x = 2) + P(x = 3) + P(x = 4)$$

The binomial probability formula is:

$$P(x = k) = \binom{n}{k} p^k (1-p)^{n-k}$$

Calculate individual probabilities

Using the Binomial Probability Distribution knowledge point, we compute each term:

For \(k = 2\):

$$P(x = 2) = \binom{9}{2} (0.4)^2 (0.6)^7 = 36 \times 0.16 \times 0.0279936 = 0.161243136$$

For \(k = 3\):

$$P(x = 3) = \binom{9}{3} (0.4)^3 (0.6)^6 = 84 \times 0.064 \times 0.046656 = 0.250822656$$

For \(k = 4\):

$$P(x = 4) = \binom{9}{4} (0.4)^4 (0.6)^5 = 126 \times 0.0256 \times 0.07776 = 0.250822656$$

Sum the probabilities and round

Using the Cumulative Binomial Probability knowledge point, we sum the calculated values:

$$P(2 \le x \le 4) = 0.161243136 + 0.250822656 + 0.250822656 = 0.662888448$$

Rounding to three decimal places yields:

$$P(2 \le x \le 4) \approx 0.663$$

Answer:

Consider a binomial random variable with \(n = 9\) and \(p = 0.4\). Let \(x\) be the number of successes in the sample. Evaluate the probability. (Round your answer to three decimal places.)

\(P(2 \le x \le 4) =\) <blank>0.663</blank>