Question

1)You own 7 Pizza Company franchises, which you acquired one by one. The managers at these franchises have been with you through thick and thin since day 1. You have now made enough money to give them bonuses, but you want to give them in such a way that the manager from Store 1 gets the most and the manager from Store 7 gets the least. You have put their bonuses in envelopes and asked your new assistant to label them. He randomly put the name of a manager on each envelope. What is the probability that the envelopes were labeled correctly?

2)

You are at a motorcycle club interviewing members about whether they wear helmets, among other things. Unknown to you, only 12 of the 20 club members present wear helmets.

What is the probability that all of the seven members you interview wear helmets? (Enter your probability as a fraction.)

(b)

How many ways can the envelopes be labeled?

(c)

You had the envelopes ordered with the one with the most money on top and the least money on the bottom. However, your soon-to-be-fired assistant dropped the first 4 envelopes and put them randomly on the top of the pile before labeling them. What is the probability that the envelopes will be in the correct money order? (Enter your probability as a fraction.)

Answer 1

Probability = Favorable Outcomes / Total Outcomes

1) (a) There are 7! ways of labeling the envelopes. 7! = 5040

(b) The favorable outcomes for the correct names to be labeled is = 1

Therefore the probability = 1 / 5040

(c) Since only 4 were dropped, the last 3 are in correct order. The 4 can be put in 4! ways = 24 ways

Favorable outcomes = Correct order = 1

Therefore the required probability = 1 / 24

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2) P(Wearing a helmet) = 12 / 20 = 3 / 5

P(Not wearing a helmet) = 1 - (3/5) = 2/5

using Binomial probability, that all 7 wear a helmet

Therefore P(X = 7) = 7C7 * (3 / 5)⁷ * (2/5)⁰ = 1 * (2187 / 78125) * 1 = 2187 / 78125

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