Problem
Adeline works at a doggie day care. The day care has 10 clients. Adeline is worried that all 10 clients will drop off their dog on Thursday. The day care owner claims that the probability that all 10 clients drop off their dogs on Thursday is less than 0.01 . Based on past Thursdays, Adeline made the following observations: - 7 of the clients drop off their dog on Thursdays $80 \%$ of the time. - 3 clients drop off their dog on Thursdays $40 \%$ of the time. Part A The event that a client drops off their dog on Thursday is independent of the event that another client drops off their dog on Thursday. If Adeline's observations are accurate, what is the probability that all 10 clients drop off their dog on Thursday? Round the answer to the nearest thousandth. Enter the answer in the box. Part B Is the day care owner's claim true or false? A. true B. false
Solution
Step 1 ：Calculate the probability for 7 clients who drop off their dogs 80% of the time: \(0.8^7 = 0.2097152\)
Step 2 ：Calculate the probability for 3 clients who drop off their dogs 40% of the time: \(0.4^3 = 0.064\)
Step 3 ：Multiply the probabilities for all 10 clients: \(0.2097152 * 0.064 = 0.0134217728\)
Step 4 ：Round the answer to the nearest thousandth: \(0.013\)
Step 5 ：\(\boxed{\text{Final Answer: Part A: 0.013, Part B: true}}\)
