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A student guesses at all 5 questions on a true-false quiz. Find each probability.

$P(\text { at least } 3 \text { correct })$

$\frac{3}{16}$

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Missouri State University

Campbell University

Oregon State University

University of Michigan - Ann Arbor

All right, So this question is asking us to find the probability of giving at least three of the five questions correct, Meaning we're not gonna be able to solve this out all at one time, because if we get three of the questions right in this scenario, that counts as a success. But if we get four questions correct, that also counts to success. If we get all five correct, that would also count is a success. So because in this scenario it's not giving us an exact situation. It's just saying, Hey, if you get three, if you get four or if you get five questions right, that all works, we're gonna have to solve out for the probabilities of each of those three scenarios, uh, individually. And then we can add them up for our total answer to this problem. Meaning first thing we'll do as we're gonna figure out the probability of getting three. Correct. Okay, let's do that. Let's figure out the probability of getting three correct. So that would mean we're doing a combination of, uh, five comma three. All right, we're doing five. Choose three. Then we're taking that. Times are probability, which we've already established is 1/2. And that would be to the third power this time because that's how many successes we had. Ah, and then we'll be taking that times the probability of us failing, which is also 1/2. And that would be of five minus three, which is gonna be to write meaning, uh, five choose three would give us 10 Epsom still in red. Five choose three would give us 10 1/2 to the third would be 1/8 and five minus three is to 1/2 squared Would be won over four. Right, Meaning we would have 10/32. That's our probability. If we get three, correct, that's great. That is not our final answer. We also need to figure out what the probability is of less getting four. Correct. So the difference would be Now we're gonna be doing five. Choose four. Still going to be a probability of 1/2 right. The probability P and Q does not change. There's still a 1/2 chance of us getting the answer. Correct. There's still a 1/2 chance of us getting the answer wrong because the 50 50 shot either way So this time it'll just be 1/2 to the fourth. And then there would be times 1/2 to the five minus four, which in this case is just gonna be one then, right, So five choose four would be five. Remember, you can do that on your calculator, right? That's not me. Just knowing that on top of my head, you're doing that in calculator. Um, 1/2 to the fourth would be 1/16 1/2 to the first, because five months for his one would just be 1/2. So this would give us the probability of 5/32. That's what the chances are that we get four. Correct. Which if you have done number 32 on this review, you already know that. Then we finally need to find the probability that we would get five correct, which we did. 31. You are. You know the answer. This is Well, so do you recognize if you're working through all these problems, you've already got some of these. But I'm gonna solve that as if you haven't done them for. So if we don't find the probability of getting five, correct that means you would be doing five. Choose five this time, and then it would be did not change colors. Then it would be the probability of a success brought, believe us, getting it right, which is 1/2 to the fifth power times the chances, the probability of us getting it wrong, which is also 1/2. But then this is going to be taken to the five minus five and minus X, which is just zero five choose. Five is one 1/2 to the fifth would be won over 32 five minus 50 Anything to the zero power is one one times 1/32 times. One would be won over 32 meaning to recap everything that we've done there. We have found that the probability of us getting three of the five questions right is 10 out of 32. The probability of us getting four of the questions right is five out of 32 and the probability of us getting five questions correct is one out of 32. So if I want to know what the probability is of getting at least three meaning 34 or five, correct, I need to take 10/32 plus 5/32 plus 1/32. Remember, when we add fractions, we need a common denominator, but we do have one here, so that's great. So then 10 plus five is 15. 15 plus one is 16 so 16/32 would be our probability, which can simplify to 1/2. So the probability of us getting at least three correct is 1/2 real quick here. I do wanna point out this is the way I would want you guys to do it because this would have worked no matter what they had asked for. But in this specific scenario, there wasn't easier way to figure this out, because think about what the options are. You can get zero questions, correct. You can get one question. Correct. You can get two questions. Correct. 34 or five. Correct. Right? Those are the six different possible outcomes that you could get. So we were looking for getting three or more, meaning we're looking to get at least three correct, and what we couldn't have happen was getting less than three. So if we really just thought about it, There were six different scenarios and we were allowed three of them, so we could have known the probability was exactly 1/2. But using the binomial distribution is much better. Because that way, if the next question asked of you was the probability of getting at least to correct, you can still solve this out. You just need to do the probability of two and add that in right.

University of Central Missouri