Arrivals to a Car Wash. Cars arrive at a car wash randomly...

Name: Problem 69, Chapter 5: Essentials of Statistics for Business & Economics
Uploaded: 2019-11-02T23:46:34-08:00
Duration: 3 min 14 s
Channel: Andrew Kim
Description: Arrivals to a Car Wash. Cars arrive at a car wash randomly and independently; the probability of an arrival is the same for any two time intervals of equal length. The mean arrival rate is 15 cars per hour. What is the probability that 20 or more cars will arrive during any given hour of operation?

Arrivals to a Car Wash. Cars arrive at a car wash randomly and independently; the probability of an arrival is the same for any two time intervals of equal length. The mean arrival rate is 15 cars per hour. What is the probability that 20 or more cars will arrive during any given hour of operation?

Key Concepts

Poisson Process

A Poisson process is a statistical model used to describe events that occur independently and at a constant average rate over time. It is commonly applied in scenarios where the timing of events (such as arrivals) is random but can be characterized by a mean rate, making the assumption that occurrences in disjoint time intervals are independent.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed period of time, given that these events happen with a fixed mean rate and independently of the time since the last event. It is used to calculate the probability of a given number of events, such as the number of cars arriving at a car wash in an hour.

Complement Rule for Cumulative Probability

The complement rule in probability states that the probability of an event occurring is one minus the probability of it not occurring. In the context of the Poisson distribution, this rule is particularly useful for calculating tail probabilities, such as the probability of having a number of events greater than or equal to some threshold, by subtracting the cumulative probability of fewer events from one.

Rate Parameter (?)

The rate parameter, denoted as ? (lambda), represents the average number of occurrences or arrivals in a given time interval in a Poisson process. This parameter defines the shape of the Poisson distribution and is fundamental to calculating event probabilities over specified intervals.

Transcript

0:00 All right.

00:01 So we're given that cars arrive at a car wash independently of one another at random intervals.

00:06 And we're given that the probability that a car arrives in one hour does not change the probability of a car arriving in the next hour.

00:15 Not sure how realistic that is, but for the sake of the problem, let's assume that.

00:19 And we're given that there's a mean number of 15 cars per hour coming through this car wash.

00:26 We want to find the probability of at least 20 cars going through the car wash in a certain hour.

00:33 Now, the reason i'm doing this in excel, instead of doing it by hand, is a, the textbook teaches you how to use excel, so i feel like it's apt.

00:41 And also, it's going to be a lot faster to use excel than explain everything and write out everything, et cetera, because we have to add potentially 20 terms.

00:51 So what i'm going to do is let's just set this equal to zero for now because we are eventually going to have to look at all the numbers between zero and 19 i'll get to that in the second so how we're going to do this problem is that the probability that at least 20 cars come through is one minus the probability that less than 20 cars come through so how we're going to do that is that we're going to use a spreadsheet to calculate all this for us i assume that no timed situation like a test would have you add up and calculate this many terms.

01:27 So no harm, no foul.

01:29 So this is a poisson distribution, so let's use that to our advantage.

01:32 So we're given that our mean rate is 15 per hour, and we're looking for an hour, so we don't need to worry about that also.

01:38 It's going to be 15 to.

01:40 This a column is going to be our column x's multiplied by.

01:46 We use exp to mean e to a certain number, e to z...

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