Question

dialsd-algebra ii, s2-rang
binomial distribution
pre-test complete
time remaining
53:08
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mrs. gomes found that 40% of students at her high school take chemistry. she randomly surveys 12 students. what is the probability that exactly 4 students have taken chemistry? round the answer to the nearest thousandth.
$p(k\text{ successes})=_{n}c_{k}p^{k}(1-p)^{n-k}$
$_{n}c_{k}=\frac{n!}{(n-k)!\cdot k!}$
0.005
0.008
0.213
0.227

Explanation:

Step1: Identify given values

$n=12$, $k=4$, $p=0.4$, $1-p=0.6$

Step2: Calculate combination $_nC_k$

$$ _{12}C_4 = \frac{12!}{(12-4)! \cdot 4!} = \frac{12!}{8! \cdot 4!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 $$

Step3: Calculate $p^k$ and $(1-p)^{n-k}$

$p^k = 0.4^4 = 0.0256$
$(1-p)^{n-k} = 0.6^{8} = 0.01679616$

Step4: Compute final probability

$$ P(4) = 495 \times 0.0256 \times 0.01679616 $$

$$ P(4) = 495 \times 0.000430984704 = 0.21333742848 $$

Step5: Round to nearest thousandth

$P(4) \approx 0.213$

Answer:

0.213