A binomial probability experiment is conducted with the given parameters. Compute the probability of successes in the in independent frials of the experiment

(Dolnot round untit the final answer. Then round to four decimal places as needed.

Ask by Santiago Christensen. in the United States

Mar 26,2025

Question

A binomial probability experiment is conducted with the given parameters. Compute the probability of successes in the in independent frials of the experiment

(Dolnot round untit the final answer. Then round to four decimal places as needed.

Ask by Santiago Christensen. in the United States

Mar 26,2025

UpStudy AI · Accepted Answer

To calculate the probability of $$ x $$ successes in $$ n $$ independent trials of a binomial probability experiment, we can use the binomial probability formula:

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

Given:
- $$ n = 15 $$ (number of trials)
- $$ p = 0.8 $$ (probability of success in each trial)
- $$ x = 13 $$ (number of successes)

Substitute the given values into the formula:

$$ P(13) = \binom{15}{13} (0.8)^{13} (1-0.8)^{15-13} $$

Now, we can calculate the probability using the binomial probability formula.
Calculate the value by following steps:
- step0: Calculate:
  $$ { }_{15}C_{13}\times 0.8^{13}\left(1-0.8\right)^{2}$$
- step1: Subtract the numbers:
  $$ { }_{15}C_{13}\times 0.8^{13}\times 0.2^{2}$$
- step2: Expand the expression:
  $$\frac{15!}{13!\times \left(15-13\right)!}\times 0.8^{13}\times 0.2^{2}$$
- step3: Subtract the numbers:
  $$\frac{15!}{13!\times 2!}\times 0.8^{13}\times 0.2^{2}$$
- step4: Reduce the fraction:
  $$\frac{15\times 14}{2!}\times 0.8^{13}\times 0.2^{2}$$
- step5: Multiply the numbers:
  $$\frac{210}{2!}\times 0.8^{13}\times 0.2^{2}$$
- step6: Divide the terms:
  $$105\times 0.8^{13}\times 0.2^{2}$$
- step7: Convert the expressions:
  $$105\left(\frac{4}{5}\right)^{13}\times 0.2^{2}$$
- step8: Convert the expressions:
  $$105\left(\frac{4}{5}\right)^{13}\left(\frac{1}{5}\right)^{2}$$
- step9: Simplify:
  $$105\times \frac{4^{13}}{5^{13}}\times \left(\frac{1}{5}\right)^{2}$$
- step10: Multiply the terms:
  $$\frac{21\times 4^{13}}{5^{12}}\times \left(\frac{1}{5}\right)^{2}$$
- step11: Evaluate the power:
  $$\frac{21\times 4^{13}}{5^{12}}\times \frac{1}{5^{2}}$$
- step12: Multiply the fractions:
  $$\frac{21\times 4^{13}}{5^{12}\times 5^{2}}$$
- step13: Multiply:
  $$\frac{21\times 4^{13}}{5^{14}}$$
The probability of $$ x = 13 $$ successes in $$ n = 15 $$ independent trials of the experiment is approximately $$ 0.230897 $$ when rounded to four decimal places.
The probability of 13 successes in 15 trials is approximately 0.2309.

A binomial probability experiment is conducted with the given parameters. Compute the probability of successes in the in independent frials of the experiment

(Dolnot round untit the final answer. Then round to four decimal places as needed.

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A binomial probability experiment is conducted with the given parameters. Compute the probability of successes in the in independent frials of the experiment (Dolnot round untit the final answer. Then round to four decimal places as needed.

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A binomial probability experiment is conducted with the given parameters. Compute the probability of successes in the in independent frials of the experiment

(Dolnot round untit the final answer. Then round to four decimal places as needed.