An online magazine finds that the mean number of typographical...

Question

An online magazine finds that the mean number of typographical errors per page is five. Find the probability that the number of typographical errors found on any given page is (a) exactly five, (b) less than five, and (c) exactly zero.

Accepted Answer

Welcome back everyone to another video. A call center receives an average of 3 customer complaints per hour. Assuming the number of complaints follows up was on distribution. Find the probability that in a randomly selected hour there are 1, exactly 3 complaints, 2, fewer than 3 complaints, and number 3, no complaints at all. Let we call the Poisson probability distribution formula. The probability of a random variable x being equal to a lower case x is equal toe raises the power of negative lambda multiplied by lambda raises the power of x divided by x factorial. We're going to use this formula to solve all three parts. Number 1, we have exactly 3 complaints, which means that our random variable X, which is the number of complaints, is equal to 3. So using the formula, we take E, we raise it to the power of negative lambda, and our lambda is 3 because this is the average or the expected value. So E raises the power of -3, multiplied by lambda raises the power of X. So 3 raises the power of 3. Divided by 3 factorial. So now we can use a calculator to evaluate this result. We end up with 0.2240, and that's our first answer. For number 2, we want to calculate the probability that we have fewer than 3 complaints. In other words, the probability of X being less than 3. So what are the possibilities? We can have 0 complaints, the probability of X being 0. Using the addition rule because we use or. We can add the probability that X is 1, and finally the probability that X is 2. Those are all of the possible events for X being less than 3, knowing that we're considering the number of complaints, right. So we start from 0, followed by 1, and then finally we have 2. What we want to do is simply apply the formula for each of those probabilities. Starting with X equals 0, we take E to the power of -3, multiplied by 3 raises the power of 0 divided by 0 factorial. We then add E, raise the power of -3, because once again, this is our lambda, multiplied by 3, raise the power of 1, we take our X. Divided by one factorial. And finally As we can see, according to the pattern, we're adding E to the power of -3 multiplied by 32, divided by 2 factorial. We can perform the calculation. And we end up with a 0.4232. And finally for part number 3. No complaints at all. The probability that X is 0. So that would be our first term from part number 2. E raises the power of -3, multiplied by 3 raises the power of 0 divided by 0 factorial, or basically Eas to the power of -3, which is equal to 0.0498. Well then we have our answers. Let's label them. And thank you for watching.

An online magazine finds that the mean number of typographical errors per page is five. Find the probability that the number of typographical errors found on any given page is (a) exactly five, (b) less than five, and (c) exactly zero.

Key Concepts

Poisson Distribution

Mean (λ) in Poisson Distribution

Probability Calculation

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