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Problem 32 Hard Difficulty

A student guesses at all 5 questions on a true-false quiz. Find each probability.
$P(\text { exactly } 4 \text { correct })$

Answer

$\frac{5}{32}$

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Top Algebra Educators
Heather Z.

Oregon State University

Kayleah T.

Harvey Mudd College

Caleb E.

Baylor University

Michael J.

Idaho State University

Video Transcript

All right. So we've got ourselves another binomial distribution problem here. It's asking us again if we take a five question. True false quiz. What is the probability? Where the chances that we could get exactly four. Correct, if we guess on every problem. Okay, so, just like before, this is our binomial distribution again. This is a review from 12-8. So you should have already seen this. If you haven't seen this thing, you should back up and go through the actual section because it's not meant to be you learning it for the first time. This supposed be reviewing. So it's assumed that you've already seen this binomial distribution before. Okay, so we know there's four variables that we need to be able to, um, plug everything in to actually solve out this formula. We need the number of total trials. We need the number of successes that we're looking for. We need the probability that we will get a success on any given trial and the probability that we will fail on any given trial. Okay, well, we've been told this is a five question test, right? So that is five different trials. Five separate questions that we have a chance to get right. We are being asked specifically on this question the probability of us getting exactly four correct. So that's the number of successes that we want. We want four successes. Okay, so now the question is what is the probability on either on any question that we will succeed versus the probability on any question that we will fail? That's the nice part about this. Question is, it's very straightforward because it's a true false test. There's only two options. So every question when we take a guess, we have a 50 50 shot that we're going to get it right. Meaning there is a one in the wrong color. There is a one in two chance that we will get it right and there is a one in two chance that we will get it wrong because there's two possible answers. We either pick true or re pick false, and one of them is right. One of them is wrong, so one out of two chance for each of them. Okay, as long as we understand all of that, the rest of this is just plugging into the formula, plugging it into our calculator and we should be done. So if we plug it into the formula up there, we've got P of X equals the combination of five hopes change the color combination of five comma four. So five choosing four times the probability of success which is 1/2 being taken to the power of how many successes we want, which we wanted four successes. Then we have to multiply that by the probability that we will fail, which is also 1/2 1 out of two. But that is being taken to the power of our number of trials minus our number of successes, which would be five minus four. Now, if you go to your calculator, you should have a button somewhere that looks something like this, and that should allow you to do combinations. Hopefully, you've already found that cause again, the ideas. You've already done this section before, So you should have already had to plug into your calculator like this before. Might have to be in a menu, But you should able to find that button somewhere. So that will allow you to do five choose for which should give you five. Then we've got 1/2 to the fourth, Power one after the fourth. Power is 1/16 because one of the fourth is one and to the 4th 2 times two is 44 times to his 88 times two is 16. All right, five minus four. For that, exponents would give us 1/2 to the first power. Anything to the first power is just itself. Meaning we've got five, which maybe we should really write as a fraction since everything else is a fraction. So remember, any whole number can be expressed as a fraction by putting it over one. So we'll just write. Five is 5/1 times 1/16. Was it really bad? Six and 1/2 to the first. Power is just 1/2. So now we just need to multiply all of these together and we've got our answer. Remember, when you are multiplying fractions, you just multiply. All the numerator is together. Multiply all the denominators together. So five times one times one, his five one time 16 times to would be 32. Meaning that our chances of getting four out of the five questions on a true false test correct if we guess, is five out of 32

University of Central Missouri
Top Algebra Educators
Heather Z.

Oregon State University

Kayleah T.

Harvey Mudd College

Caleb E.

Baylor University

Michael J.

Idaho State University