Aydan Hardy

2023-03-31

A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.n=20​,
p=0.7​,
x=19
P(19)=

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To calculate the probability of having 19 successes in 20 independent trials of a binomial probability experiment, we can use the binomial probability formula:
$P\left(x\right)=\left(\genfrac{}{}{0}{}{n}{x}\right)·{p}^{x}·\left(1-p{\right)}^{n-x}$,
where $n$ represents the number of trials, $x$ is the number of successes, and $p$ is the probability of success in a single trial.
Given:
$n=20$ (number of independent trials)
$p=0.7$ (probability of success in a single trial)
$x=19$ (number of successes)
Now, let's substitute these values into the formula and calculate the probability:
$P\left(19\right)=\left(\genfrac{}{}{0}{}{20}{19}\right)·{0.7}^{19}·\left(1-0.7{\right)}^{20-19}$.
The binomial coefficient, $\left(\genfrac{}{}{0}{}{20}{19}\right)$, can be calculated as:
$\left(\genfrac{}{}{0}{}{20}{19}\right)=\frac{20!}{19!\left(20-19\right)!}$.
Simplifying this expression, we have:
$\left(\genfrac{}{}{0}{}{20}{19}\right)=\frac{20!}{19!·1!}=20$.
Now, let's substitute the calculated values into the probability formula:
$P\left(19\right)=20·{0.7}^{19}·\left(1-0.7{\right)}^{20-19}$.
Calculating the exponents:
$P\left(19\right)=20·{0.7}^{19}·{0.3}^{1}$.
Simplifying the expression:
$P\left(19\right)=20·{0.7}^{19}·0.3$.
Now, let's calculate the numerical value of this probability:
$P\left(19\right)=20·{0.7}^{19}·0.3\approx 0.00684$.
Therefore, the probability of having 19 successes in 20 independent trials of the binomial probability experiment is approximately 0.00684.

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