Questions: Find the probability of exactly three successes in eight trials of a binomial experiment in which the probability of success is 45%. n=8 p=0.45 q=0.55 k=3 Enter values into the formula. P(k)=nCk(p)^k(q)^(n-k) P(3)=8C3(0.45)^3(0.55)^5

Find the probability of exactly three successes in eight trials of a binomial experiment in which the probability of success is 45%.
n=8 p=0.45 q=0.55 k=3

Enter values into the formula.
P(k)=nCk(p)^k(q)^(n-k)
P(3)=8C3(0.45)^3(0.55)^5

Transcript text: SKIp Find the probability of exactly three successes in eight trials of a binomial experiment in which the probability of success is $45 \%$. \[ \mathrm{n}=8 \quad \mathrm{p}=0.45 \quad \mathrm{q}=0.55 \quad \mathrm{k}=3 \] Enter values into the formula. \[ \begin{array}{r} P(k)={ }_{n} C_{k}(p)^{k}(q)^{n-k} \\ P(3)={ }_{8} C_{[?]}(\square)(\square) \end{array} \]

Solution

Solution Steps

Step 1: Define the Problem

We are tasked with finding the probability of exactly $ k = 3 $ successes in $ n = 8 $ trials of a binomial experiment, where the probability of success on each trial is $ p = 0.45 $. The probability of failure is given by $ q = 1 - p = 0.55 $.

Step 2: Apply the Binomial Probability Formula

The probability of exactly $ k $ successes in $ n $ trials is given by the formula: \[ P(k) = {n \choose k} p^k q^{n-k} \] Substituting the known values: \[ P(3) = {8 \choose 3} (0.45)^3 (0.55)^{8-3} \]

Step 3: Calculate the Probability

Using the binomial probability mass function, we find: \[ P(3) \approx 0.2568 \]