There is a 40% chance that a customer will choose Diner A for takeouts and a probability of 60% that he/she will choose Diner B. If he/she orders from Diner A, 12 minutes after placing the order, the remaining time it takes for the food to be delivered is exponentially distributed with an average of 14 minutes. If she orders from Diner B, 17 minutes after placing the order, the time it takes for the food to arrive is exponentially distributed with an average of 11 minutes. Given that he/she has already waited 27 minutes after placing the order, and the food hasn't arrived yet, what is the probability that he/she ordered from Diner A? HINT: Use continuous probability distributions methods.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
There is a 40% chance that a customer will choose Diner A for takeouts and a
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