Mail Ordering A mail-order company receives an average of 5 orders per 500 solicitations. If it

Mail Ordering A mail-order company receives an average of 5...

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has two outcomes (success or failure) and the same probability of success. In this context, each advertisement represents a trial where an order (success) is placed with a certain probability, making the number of orders received a binomial random variable.

Poisson Approximation

The Poisson approximation is used when the number of trials is large and the probability of success in each trial is small. It simplifies calculations by approximating the binomial distribution with a Poisson distribution, which is characterized by its mean. This approach is particularly useful in scenarios involving rare events, such as receiving orders from advertisements when the success probability is low.

Complement Rule

The complement rule in probability states that the probability of an event happening is one minus the probability of the event not happening. In problems where it is easier to calculate the likelihood of the opposite outcome (for example, receiving fewer than two orders), the complement rule can be applied to find the probability of receiving at least two orders by subtracting the probability of receiving less than two from one.

Transcript

00:01 A pusan probability is when we are determining a probability in a defined space, a period of time, or volume.

00:09 And this particular problem is talking about finding the probability of receiving two orders in a defined number of advertisements going out, and that would be a defined space, 100 advertisements.

00:23 So let's write down what we know.

00:27 We know that there's five orders.

00:35 That are received per 500 solicitations or advertisements.

00:45 We know that it is a poucant probability, and the formula for a pusan probability is the probability of x given an average or a lambda equals e to the negative a lambda times lambda raised to the x power, all over x factorial.

01:16 And lambda in this case is the average number of orders in a hundred advertisements rather than 500.

01:27 So we're going to have to set up a proportion to solve this.

01:30 We know that there are five orders generated when we send out 500 ads or solicitations.

01:43 And we've got to figure out our lambda when we only send out 100 ads.

01:52 And if we do our cross -product, we multiply, we would end up with 500 lambda equals 500, or lambda equals 1.

02:13 So we know that we average one order for every hundred solicitations or ads that go out.

02:23 Now, we're trying to find the probability of receiving at least two orders.

02:28 So if we define what at least two orders means, well, that could mean that we have two orders or three orders or four or five...

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