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

## $\frac{1}{32}$

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so we're being asked probability question here, but specifically were being asked about five questions on a true or false quiz. Now it's extremely important here that this that we understand this is a true or false quiz. That's why they gave us the information as opposed to, you know, in multiple choice quiz or something like that to the point of them. Saying it's a true or false quiz is it means there are only two options. It's a binary decision. You're either going to get it right or you're going to get it wrong, because it does say that student is guessing. So it's not like believing any of these blank, right? It does clearly say the student guesses. So you're assuming that there is an answer put by he or she. In that case, it's a 50 50 shot every single time, right, because you're going to get it right or get it wrong. So because of that, that makes this a binomial distribution problem or a problem that we can solve using the binomial distribution because we can use this for any time. You have a binary situation, a situation where there are only two outcomes Okay. Hence the buying binomial right? That's a prefix for two. So we have our formula here, the binomial distribution which you guys should have seen already because this is the review part where it's going back to 12-8. So it is assuming that you've already done that before. It's not with this section was about. It's what nearly your section was about eso. What we've got here is that your probability for years in a row is going to be equal toothy combination of in being the number of trials and X being the number of successes. Okay, so and is like the total amount of trials that you're doing. And X is whatever you're wanting to get. Whatever the question is asking you about, then you've got P to the X. Soapy is the probability of you having success to the power of X, which we already said was the number of successes then that's times Q. Which is the probability of failure, and that's being taken to the power of and minus X. So the number of trials minus the number of successes meaning basically, how many times do you not need to succeed? Okay, So if we look at the specific problem, so we've got five questions, Okay, so right off the bat there, that's how many questions we have. That means that's our total. That's end. That's the number of trials quote unquote, that we're running because you've got five chances to get it right. Answer right for the specific problem it wants us to fund the probably that you get all five correct meaning we have five trials. That's N That would have been the same, no matter what question we were doing. As long as the scenario is five questions on the quiz that also is giving us X, though, because it's saying it wants us to get all five correct, meaning it wants us to have five successes. Okay, so then the other two variables that we need R P and cue well, p represents the probability of success. So on any given question, if you're looking at if that students looking at question one where they have true or false, what is the chance that what is the probability that will get it right? Well, you've got two answers to possible answers. One of them is correct. So that would mean he or she has a one out of two, a one in two chance of getting a question right, but they could pick the wrong one. So Q is also won over to write literally. A true false test is a 50 50 shot, so the probabilities are going to be half in half 50 50 right? So now that we have all of that, we can toss it into our binomial distribution up at the top of the page, and we should be able to get ourselves an answer. So P of X is equal to a combination of five Connor five times probability of success, which is 1/2 the P value being taken to the X power, which is five. Because that's the number of successes we want. We want to get all five correct, and then lastly, we're multiplying that to the probability of failure, which is also 1/2 because again, 50 50 shot, um and that is going to be taken to the n minus X power, which is five minus five. Because in this case, an and X are the same thing you came so solving that down for the combination You should be able to do that in your calculator. If you go look at your calculators somewhere either on your buttons or you may have to go into a menu depending on your calculator. But there should be a symbol that looks something like this. N choose R N C r looks something like that. Okay, if you find that button, you should be able to hit that and then put in five common five. If you're plugging it incorrectly, you should get one for that. The combination of five comma five and five choose five. Should be one. Okay, then you can go to your calculator and do 1/2 to the fifth. Or you might will do that in your head if you remember the rules for exponents, because remember a time you're taking a fraction to a power. It's the same is just taking the numerator and the denominator to that power separately, meaning one to the fifth power is one. And to to the fifth power is 32 meaning 1/2 to the fifth power. Is that 1/32. Okay, and then we've got 1/2 to the five minus five. Well, five, minus 50 So that means we're doing 1/2 to the zero power. Anything to the zero power is always one. Meaning what we effectively have is one times 1/32 times one. Well, one times 1/32 times, one would just be 1/32. And that would be the probability that on a true false quiz where you guess on five questions, you would get all five of those correct.

University of Central Missouri