Questions: Sandy's Pet Center grooms large and small dogs. It takes Sandy 40 minutes to groom a small dog and 70 minutes to groom a large dog. Large dogs account for 20% of Sandy's business. Sandy has 5 appointments tomorrow. Assume this example meets all the characteristics of a binomial experiment. DEFINE SUCCESS AND FAILURE: SUCCESS = FAILURE = a. What is the probability that all 5 appointments tomorrow are for small dogs? b. What is the probability that less than 2 of the appointments tomorrow are for large dogs? c. What is the expected amount of time to finish all 5 dogs tomorrow?

Solution Steps

In this scenario, we define:

• Success: Grooming a small dog.
• Failure: Grooming a large dog.

Given that large dogs account for \(20\%\) of Sandy's business, the probability of success \(p\) (grooming a small dog) is \(0.8\) and the probability of failure \(q\) (grooming a large dog) is \(0.2\).

To find the probability that all 5 appointments are for small dogs, we use the binomial probability formula:

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

Substituting \(n = 5\), \(x = 5\), \(p = 0.8\), and \(q = 0.2\):

\[ P(X = 5) = \binom{5}{5} \cdot (0.8)^5 \cdot (0.2)^0 = 1 \cdot 0.32768 \cdot 1 = 0.3277 \]

Thus, the probability that all 5 appointments are for small dogs is \(0.3277\).

To find the probability that less than 2 of the appointments are for large dogs, we calculate the probabilities for \(X = 0\) and \(X = 1\):

1. For \(X = 0\): \[ P(X = 0) = \binom{5}{0} \cdot (0.2)^0 \cdot (0.8)^5 = 1 \cdot 1 \cdot 0.32768 = 0.3277 \]

2. For \(X = 1\): \[ P(X = 1) = \binom{5}{1} \cdot (0.2)^1 \cdot (0.8)^4 = 5 \cdot 0.2 \cdot 0.4096 = 0.4096 \]

Adding these probabilities gives: \[ P(X < 2) = P(X = 0) + P(X = 1) = 0.3277 + 0.4096 = 0.7373 \]

Thus, the probability that less than 2 of the appointments are for large dogs is \(0.7373\).

To calculate the expected amount of time to finish all 5 dogs, we first find the expected number of large dogs:

\[ \mu = n \cdot p = 5 \cdot 0.2 = 1.0 \]

The expected number of small dogs is: \[ 5 - \mu = 5 - 1 = 4 \]

Next, we calculate the expected grooming time:

• Time for small dogs: \(40\) minutes each
• Time for large dogs: \(70\) minutes each

Thus, the expected time is: \[ \text{Expected Time} = (4 \cdot 40) + (1 \cdot 70) = 160 + 70 = 230 \text{ minutes} \]

Final Answer