Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to use selection with replacement to approximate selection without replacement. Here's our problem statement: In a study of helicopter usage and patient survival, among the 54,115 patients transported by helicopter, 172 of them left the treatment center against medical advice, and the other 53,943 did not leave against medical advice. If 40 of the subjects transported by helicopter are randomly selected without replacement, what is the probability that none of them left the treatment center against medical advice?
OK, we're asked for a probability, and probability is a part over the whole. But we're looking at 40 subjects here and the probability that none of them left the treatment center. So this is going to be the probability that the first one didn't leave and the probability that the second one didn't leave and the probability that the third one didn't leave, so on and so forth until we get to the 40th subject. Well, and, when we're dealing with probability, calculations means multiplication. So we're taking this probability that the randomly selected person did not leave the treatment center against medical advice and we're multiplying it by itself 40 times.
But that's when you're selecting with replacement here. Technically we would need to look at selecting without replacement, because once the person leaves, it's not like they come back into the hospital to get selected again. So we're going to have to --- if we want to do this technically correct, we will be selecting without replacement. Well, that means we're going to have to take this 53,943 and divide it by 54,115 --- there's your first probability. But then we got to multiply that by 53,942 over 54,114 to account for the one person that is no longer available to be selected. And we'd have to continue that on for 40 different numbers that we're going to multiply together. Gee, that's some pain.
What we're going to do instead is we're actually going to use selection with replacement to approximate selection without replacement. And what allows us to do this is that the number of people that we're looking at is actually within 5% of the whole. So if we take, say, the 40 subjects that we're looking at out of the total that we can select from, so you would get a number here that's way less than one --- well, 1%. So it's definitely going to be less than 5%, which means we can use selection with replacement to approximate selection without replacement.
So again, the selection with replacement, it's just going to be the part over the whole. So the 172 that left the treatment center is the part that left, but we want the probability that none of them are leaving. So we want the 53,943 that did not leave divided by the whole. So there's a probability that one randomly selected patient is not going to leave the hospital. I want this multiplied by itself 40 times for the 40 subjects that we're selecting from the --- from the total pool of candidates. So then there is our probability. I round it to three decimal places. Fantastic!
Now this value that we get here is not that different from if we were to actually go and do the actual calculation itself in Excel. And actually it wouldn't take me long to set it up in Excel. We can run through this very quickly. So if I've got the total number of patients, and I'm selecting from the 54,115, and I'm going to select one of them from the 53,943. So then the probability is going to be the part over the whole. So that's my probability for one.
And now if I just go ahead and say I'm going to take one away from you, I'm going to take one away from you, and I'm going to select the same probability here. And we wanted to do that for the 40 patients that were looking to select. So if I just bring this down, I get the 40. And there are my individual probabilities.
Now to get the final probability, I've got to multiply all of those probabilities together, and look at the number that we get. It's not appreciably different from what we saw earlier. This is 0.880395, and this is 0.880435. So we can see that, you know, even though it's not the same number, it's close enough to where we can use it as an approximation. Again, you have to be within 5% in order for that approximation of the whole, because once you leave that 5% range, then the difference between your two numbers becomes great enough to be significant. So anyway, that's why we can actually use this selection with replacement to approximate selection without replacement. It makes the calculation much, much easier.
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Frustrated with a particular MyStatLab/MyMathLab homework problem? No worries! I'm Professor Curtis, and I'm here to help.