16 A binomial probability experiment is conducted with the given parameters. Use technology to find the probability of successes in the independent trials of the experiment

(Round to four decimal places as needed.)

Question

16 A binomial probability experiment is conducted with the given parameters. Use technology to find the probability of successes in the independent trials of the experiment

(Round to four decimal places as needed.)

Ask by Chan Frank. in the United States

Mar 26,2025

UpStudy AI · Accepted Answer

Given the parameters
$$
n=9,\quad p=0.3,
$$
we want to compute
$$
P(x<4)=P(x=0)+P(x=1)+P(x=2)+P(x=3).
$$

The probability of exactly $$ k $$ successes in a binomial experiment is given by
$$
P(x=k)=\binom{n}{k}p^k(1-p)^{n-k}.
$$

**Step 1. Compute $$P(x=0)$$:**
$$
P(x=0)=\binom{9}{0}(0.3)^0(0.7)^9 = 1\cdot1\cdot(0.7)^9.
$$
Evaluating $$(0.7)^9$$:
$$
(0.7)^9\approx 0.0404.
$$

**Step 2. Compute $$P(x=1)$$:**
$$
P(x=1)=\binom{9}{1}(0.3)^1(0.7)^8 = 9\cdot0.3\cdot(0.7)^8.
$$
Calculating:
$$
(0.7)^8\approx 0.0576,\quad 9\cdot0.3=2.7,\quad 2.7\cdot0.0576\approx 0.1557.
$$

**Step 3. Compute $$P(x=2)$$:**
$$
P(x=2)=\binom{9}{2}(0.3)^2(0.7)^7 = 36\cdot0.09\cdot(0.7)^7.
$$
With
$$
(0.7)^7\approx 0.0824,\quad 36\cdot0.09=3.24,\quad 3.24\cdot0.0824\approx 0.2670.
$$

**Step 4. Compute $$P(x=3)$$:**
$$
P(x=3)=\binom{9}{3}(0.3)^3(0.7)^6 = 84\cdot0.027\cdot(0.7)^6.
$$
Since
$$
(0.7)^6=0.117649,\quad 84\cdot0.027\approx 2.268,\quad 2.268\cdot0.117649\approx 0.2666.
$$

**Step 5. Sum the probabilities:**
$$
P(x<4)=0.0404+0.1557+0.2670+0.2666\approx 0.7297.
$$

Rounding to four decimal places, we obtain:
$$
P(x<4)\approx 0.7296.
$$
The probability of having fewer than 4 successes is approximately 0.7296.

16 A binomial probability experiment is conducted with the given parameters. Use technology to find the probability of successes in the independent trials of the experiment

(Round to four decimal places as needed.)

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16 A binomial probability experiment is conducted with the given parameters. Use technology to find the probability of successes in the independent trials of the experiment (Round to four decimal places as needed.)

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16 A binomial probability experiment is conducted with the given parameters. Use technology to find the probability of successes in the independent trials of the experiment

(Round to four decimal places as needed.)