STATS Extra Credit: Probability (Ms. Jobe)

The question reads, during the years 1886 through 2000, there were an average of 8.7 tropical cyclones per year, of which an average of 5.1 becomes hurricanes. Assuming that the probability of any cyclone becoming a hurricane is independent, of what happens to any other cyclone. If there are 5 cyclones in one year, what is the probability that at least 3 become hurricanes? This is the 9th question of Practice Examination 1 of the Barron's book. Given that this is a probability problem, we must find whether or not it's asking for binomial or geometric. For binomial, There are 4 assumptions that need to be confirmed, and these 4 are that 1, each replication results in 1 or 2 possible outcomes. 2, the probability of success is the same for each application. 3, the replications are independent. And 4, the number of trials of the chance process must be fixed in advance. And altogether, this can be put into 4 letters, which are bins. And how you know that all 4 of these are assumed, you can look in the problem. It says that each hurricane is independent of each other, which knocks out this. And the number of trials is fixed. Because they give you 5 cyclones. And then a success or failure is whether or not it becomes a hurricane or does not become a hurricane. And the probability of success will be the same for each, given that they give you the averages here. Given that, we now have our 3 values we must put into the calculator. So now I'm going to find binomial CDF of 5, p-value is 0.5862, and x value is 2. And now that will give us this value, but before we finish, we have to subtract this value. By the binomial CDF to 1, which will give us 0.65845. And now let's see if this answer choice is here, and we can see that it will give us D. But before we stop, we can realize that these other selected choices are incorrect. We can see that C gives us just the probability, which is wrong, and then A is incorrect because this gives you the binomial CDF, or 1 subtracted by binomial CDF, if your x value is 3, in which our case the x value is 2. So that's it. Question 2: A plumbing contractor obtains 60% of her boiler circulators from a company whose defect rate is 0.005, and the rest from a company whose defect rate is 0.010. If a circulator is defective, what is the probability it came from the first company? And this question is quite Question 6 of Practice Examination 1 of the AP Stats Barron's book. You can tell that this is a given problem because it gives you that the first company has a defect rate of 0.005 and the other company is 0.010, and that 60% is taken from the first company. The probability of A means that the circulator came from the first company, and given that the circulator is defective. So this would mean the probability of it being from the first company while also being defective over the probability that the circulator is defective. So in order to find the probability of the circulator being both from the first company and also being defective is given within the problem. So it says that 60% of Herb Boyle circulators is from a company whose defect rate is 0.005. So the probability of the boiler circulator being from the first company is 60%, and this defect rate is 0.005. So now you must find the probability of the circulator being defective. So within the problem, They give you one company having a defect rate of 0.005, 60% of the boiler circulators bought, and they also tell you that the rest bought is from a company whose defect rate is 0.010. So that gives you separate defect rates. So the probability of the boiler circulator being defective would be again 60% coming from the first company, which has a defect rate of 0.005. And now you must add the remaining values. So since 60% is from the first company, that means 40% is from the rest, and that's defect rate is 0.010. So this will be your answer. So now let's check back on the answer choices. So we can see that the answer will be E. So first of all, we can eliminate the first, A, because this only gives you the numerator of the actual answer. We can eliminate B because this is only the denominator of the answer, which does not follow this equation. And we can eliminate C because it only gives us the probability of 60% and not given, that is, um, that is giving us 60%, which doesn't give us— which 60% and it needs to be multiplied by the defect rate since that question is asking for the probability came from the first company given that the circulator is defective. So we can eliminate D as well because it gives us the defect rate, but it needs to be multiplied by 60%. And now we can officially say that E is the correct answer. And that's it.