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In our final lesson, we’ll talk about some guessing strategies you can use. So of course, ideally you’ll learn all the rules of probability and be able to solve all the problems. What if you get stuck at a problem? If you don’t even know how to begin, what are some ways you can guess effectively on a probability problem?
So first of all, use you instincts, and as you work the probability problems over time you will develop instincts. So for example, if it’s something like flipping a coin ten times and getting eight heads. Something like’s that, not going to be very, very likely and so if one of the answer choices is like point eight or point nine.
You know, that an event like that is not gonna happen 80% or 90% of the time. So, something like that would simply be unreasonable and you can eliminate it. So, always use your real world instincts about would something like this be really likely, really unlikely and if you can use that information, use that to eliminate some answer choices.
Second, if the complement rule is clearly used in a problem, even if you don’t know how to solve the problem, look for complementary probabilities in the answers. Now, what do I mean by this? Suppose we have this question. If blah, blah, what is the probably that at least one blah, blah, blah, blah, blah?
So we have no idea what this question is about. Suppose you look at the question and it might as well be written in blah, blah. You have no idea what the question is actually asking. But you spot these magic words, at least one, and you remember, oh, in an at least one problem, the quick way to do it is to use the Complement Rule. So of course, that’s what the test maker has in mind.
Now let’s think about this. If the test maker wants a probability of A. Of course, the way that you figure this out is it’s one minus the probability of not A. Well of course, sometimes people go through all the work of figuring out the probability of, of not A and they forget this final step of subtracting it from one.
This is a very common mistake, because this is a common mistake. It’s always true that on a problem with a compliment rule is used, even though this is the correct answer, this will also appear as a trap answer. The test maker always will put the likely mistake as a trap answer. So that means, if we see pairs of answers that are compliments, that’s a very good clue that those, one of those is the correct answer.
So for example, we look at the answer choices here. These two answers are complements, the other answers don’t have their complement listed. So it’s very likely, that those are not the answers, and we can guess from one of those two choices. And of course, if you can eliminate three choices and guess from the remaining two, your odds are much, much higher and it’s to your advantage to do so.
Finally, think about overlap. And this is a very tricky issue about probability. Suppose for example the probability of A is point 7 the probability of B is point 8 and we’re asking them a question about A and B, A or B something like this. We’ll notice first of all. If one thing is happened 70% of the time and one thing has happened 80% of the time, there is absolutely no way, that they could be mutually exclusive.
Because there’s no way that you could have 70% of the time and 80% of the time, never touching each other. They’re gonna have to overlap. One way to, to view this is visually. So for example, if this is the 80% of the time, so this would be the whole space that would be 1, this would be B which is point 8 and so that just leaves a little point 2 outside.
Suppose we want to have them have minimum overlap. Well minimum overlap, then we push A all the way over here, point 7 and so that means, it be point 3 on the outside of our this side and that means there overlap zone would be point 5 and so it means, that if we have a minimum of overlap. Notice that what we have here is that the probability of A or B equals 1.
The probability of A and B equals point 5. So we’re trying to make the overlap a minimum. What if we try and make the overlap a maximum? Well then we’ll kind of push everything to the same side. Here’s our 1. Here’s our B, we’ll push it to the left side.
That’s our point 8, and we’ll push A to the left side also. And it turns out that first of all, the or region is just the probability of being now. The probability of A or B because A is inside B, and so that’s the probability of or.
And the probability of and will now that’s the size of A probability of A and B everywhere where A is B is also. So that’s point 7. So notice, even if we don’t know anything about the particulars of the situation, just thinking about overlap. We know that probability of or has to be somewhere in this range and the probability of and has to be somewhere in this range.
So that would allow us to eliminate answer choices. In the rare case that we had an inequality question about probability, we might be able to answer the question purely by thinking about this overlapping. These are some strategies you can use even if you don’t understand the question, there’s still effective ways to eliminate the answers, so that you can guess effectively on these problems.
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