Question

evaluate the binomial probability. (round your answer to three decimal places.)

\\ c_{4}^{11} (0.5)^4 (0.5)^7 \\

Explanation:

Identify the given expression

Using the Binomial Probability Distribution knowledge point
We are given the following expression to evaluate:

$$ P = C_{4}^{11} (0.5)^4 (0.5)^7 $$

Our goal is to calculate this value and round the final result to three decimal places.

Calculate the binomial coefficient

Using the Binomial Probability Distribution knowledge point
The binomial coefficient \(C_{4}^{11}\) (also written as \(\binom{11}{4}\)) is calculated as:

$$ C_{4}^{11} = \frac{11!}{4!(11-4)!} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} $$

$$ C_{4}^{11} = \frac{7920}{24} = 330 $$

Simplify the exponential terms

Using the Binomial Probability Distribution knowledge point
Combine the powers of \(0.5\):

$$ (0.5)^4 \times (0.5)^7 = (0.5)^{4+7} = (0.5)^{11} $$

Calculate the value of \((0.5)^{11}\):

$$ (0.5)^{11} = \frac{1}{2^{11}} = \frac{1}{2048} \approx 0.00048828125 $$

Calculate the final probability

Using the Binomial Probability Distribution knowledge point
Multiply the binomial coefficient by the simplified exponential term:

$$ P = 330 \times \frac{1}{2048} = \frac{330}{2048} = \frac{165}{1024} $$

$$ P \approx 0.1611328125 $$

Round to three decimal places

Using the Binomial Probability Distribution knowledge point
Rounding \(0.1611328125\) to three decimal places:

$$ P \approx 0.161 $$

Answer:

0.161