Questions: A binomial experiment has the given number of trials n and the given success probability p. n=20, p=0.9 (a) Determine the probability P (More than 18 ). Round the answer to at least three decimal places. P( More than 18)=

A binomial experiment has the given number of trials n and the given success probability p.

n=20, p=0.9

(a) Determine the probability P (More than 18 ). Round the answer to at least three decimal places.
P( More than 18)=

Transcript text: A binomial experiment has the given number of trials $n$ and the given success probability $p$. \[ n=20, p=0.9 \] (a) Determine the probability $P$ (More than 18 ). Round the answer to at least three decimal places. $P($ More than 18$)=$ $\square$

Solution

Solution Steps

Step 1: Define the Problem

We are given a binomial experiment with $ n = 20 $ trials and a success probability $ p = 0.9 $. We need to determine the probability of obtaining more than 18 successes, which can be expressed mathematically as:

\[ P(X > 18) = P(X = 19) + P(X = 20) \]

Step 2: Calculate $ P(X = 19) $

Using the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

where $ q = 1 - p = 0.1 $, we calculate $ P(X = 19) $:

\[ P(X = 19) = \binom{20}{19} \cdot (0.9)^{19} \cdot (0.1)^{1} = 0.27 \]

Step 3: Calculate $ P(X = 20) $

Next, we calculate $ P(X = 20) $:

\[ P(X = 20) = \binom{20}{20} \cdot (0.9)^{20} \cdot (0.1)^{0} = 0.122 \]

Step 4: Calculate $ P(X > 18) $

Now, we sum the probabilities of exactly 19 and 20 successes to find $ P(X > 18) $:

\[ P(X > 18) = P(X = 19) + P(X = 20) = 0.27 + 0.122 = 0.392 \]