Find the indicated probabilities using the geometric...

Find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities. The mean number of oil tankers at a port city is eight per day. Find the probability that the number of oil tankers on any given day is (a) exactly eight, (b) at most three, and (c) more than eight.

Find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.
The mean number of oil tankers at a port city is eight per day. Find the probability that the number of oil tankers on any given day is
(a) exactly eight, (b) at most three, and (c) more than eight.

Key Concepts

Poisson Distribution

The Poisson distribution is a discrete probability distribution that is often used to model the number of times an event occurs in a fixed interval of time or space. It is particularly useful when the events occur independently at a constant rate, as given by the parameter ? (lambda), which represents the average number of events per interval.

Parameter ? (Lambda) and Its Role

The parameter ? in the Poisson distribution is the expected number of events in the specified interval. In problems like these, ? is used in the probability mass function (PMF) and determines the shape of the distribution. Knowing ? allows you to calculate probabilities of different outcomes, such as exactly eight events or at most three events.

Probability Mass Function (PMF)

The PMF of the Poisson distribution is used to determine the probability of observing exactly a specific number of events. It is given by the formula P(X = x) = (e^(??) * ?^x) / x! for x = 0, 1, 2, …. This formula is key to solving parts of the problem that ask for the probability of exactly eight events.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) for the Poisson distribution is used to calculate the probability of observing a number of events less than or equal to a specific value. This is achieved by summing the PMF values for all events up to that value, which is useful when answering parts of the problem that ask for probabilities of 'at most' a certain number of events or 'more than' a certain number.

Assessing Unusual Events

Determining whether an event is unusual involves comparing the calculated probability to a predetermined threshold (commonly 0.05). If the probability of an event is very low—below this threshold—the event is generally considered unusual. This concept helps in evaluating the significance of the outcomes in probabilistic problems.

Transcript

00:01 So we know that the mean number of tankers that come into this area is eight per day.

00:07 And we want to know what's the likelihood that exactly eight come in in a day? what's the likelihood that three or fewer come in in a day? and what's the likelihood that more than eight come in a day? and this is the poisson setting.

00:29 And so since we're allowed to use the text, features.

00:33 Let's use the technology features.

00:35 So we know that this is a poisson distribution and we're going to use a pdf for this first one and we know that the mean is eight and the number that we want to have as far as how many occur is what the mean is.

00:50 So we go and use our poisson distribution pdf for a particular value and eight and eight and that comes out to be 0 .13.

01:03 Nine five will say nine so about 14 % chance now what's the likelihood that three or fewer come in in that day so zero one two three well we want to use our poison cedia for that because that's going to end up being four different events and we know the mean is eight and we know that we want three or fewer and make sure you use that accumulative button and so nice that this is programmed this way so again our mean is eight and our number is 3 for the x value.

01:38 And when we do that calculation, we find out that that value comes up to be about that.

01:45 So that's not real likely to happen.

01:47 Now, what's the likelihood that more than 8? more than 8, we're going to have to find 1 minus and we'll have the poisson cumulative button, cdf...

Please give Ace some feedback