Questions: A market analysis team selects 7 random volunteers to taste 5 Bacon Cheeseburgers from local restaurants. 4 from the national chain and 1 from a local diner. a. What is the probability that exactly 3 people will pick the local diner's burger? b. What is the probability that none of the people select the local diner's burger?

Solution Steps

We are tasked with finding the probabilities related to a group of 7 volunteers tasting 5 Bacon Cheeseburgers, where 4 are from a national chain and 1 is from a local diner. We need to calculate:

1. The probability that exactly 3 people will pick the local diner's burger.
2. The probability that none of the people select the local diner's burger.

To find the probability that exactly 3 out of 7 volunteers select the local diner's burger, we use the binomial probability formula:

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

Where:

• \( n = 7 \) (total volunteers)
• \( x = 3 \) (volunteers picking the local burger)
• \( p = \frac{1}{5} \) (probability of picking the local burger)
• \( q = \frac{4}{5} \) (probability of not picking the local burger)

Substituting the values, we find:

\[ P(X = 3) = \binom{7}{3} \cdot \left(\frac{1}{5}\right)^3 \cdot \left(\frac{4}{5}\right)^{4} = 0.1147 \]

Thus, the probability that exactly 3 people will pick the local diner's burger is \( 0.1147 \).

Next, we calculate the probability that none of the 7 volunteers select the local diner's burger. Again, we use the binomial probability formula:

\[ P(X = 0) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

Where:

• \( n = 7 \)
• \( x = 0 \)
• \( p = \frac{1}{5} \)
• \( q = \frac{4}{5} \)

Substituting the values, we find:

\[ P(X = 0) = \binom{7}{0} \cdot \left(\frac{1}{5}\right)^0 \cdot \left(\frac{4}{5}\right)^{7} = 0.2097 \]

Thus, the probability that none of the people select the local diner's burger is \( 0.2097 \).

Final Answer