Question

assume that a procedure yields a binomial distribution with a trial repeated n times. use the binomial probability formula to find the probability of x successes given the probability p of success on a single trial. round to three decimal places.
$n=10, x=4, p=0.25$
○ 0.179
○ 0.128
○ 0.146

Explanation:

Step1: Recall binomial probability formula

The binomial probability formula is:
$$P(X=x) = \binom{n}{x} p^x (1-p)^{n-x}$$
where $\binom{n}{x} = \frac{n!}{x!(n-x)!}$

Step2: Calculate combination term

For $n=10, x=4$:
$$\binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10\times9\times8\times7}{4\times3\times2\times1} = 210$$

Step3: Compute $p^x$ and $(1-p)^{n-x}$

$p=0.25$, so:
$$p^x = 0.25^4 = 0.00390625$$
$$(1-p)^{n-x} = (0.75)^{6} = 0.1779785156$$

Step4: Multiply all terms together

$$P(X=4) = 210 \times 0.00390625 \times 0.1779785156$$
$$P(X=4) = 210 \times 0.00069580078125 = 0.1461181640625$$

Step5: Round to 3 decimal places

$$P(X=4) \approx 0.146$$

Answer:

0.146