The probability of a customer arrival at a grocery service...

The probability of a customer arrival at a grocery service counter in any one second is equal to. $1 .$ Assume that customers arrive in a random stream and hence that an arrival in any one second is independent of all others. Find the probability that the first arrival a. will occur during the third one-second interval. b. will not occur until at least the third one-second interval.

The probability of a customer arrival at a grocery service counter in any one second is equal to. $1 .$ Assume that customers arrive in a random stream and hence that an arrival in any one second is independent of all others. Find the probability that the first arrival a. will occur during the third one-second interval.
b. will not occur until at least the third one-second interval.

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Key Concepts

Geometric Distribution

This distribution models the number of independent trials needed to achieve the first success where each trial has the same probability of success. In problems involving the waiting time until the first occurrence of an event (like a customer arriving), the geometric distribution is used to calculate probabilities such as the event occurring exactly on a specified trial (e.g., the third one?second interval) or not occurring before a certain trial.

Independent Bernoulli Trials

The problem assumes that customer arrivals in each one-second interval occur independently of arrivals in other intervals. This means that each second can be treated as a Bernoulli trial with only two outcomes (arrival or no arrival), and the probability of each outcome remains constant across intervals.

Memoryless Property

A key characteristic of the geometric distribution is that it is memoryless. This means that the probability of an event occurring in the future is independent of the past; for instance, if no arrival has occurred in the first two intervals, the chance of an arrival in the next second is unaffected by those previous intervals.

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