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So in previous videos, we have talked about all these ideas. Mutually exclusive, independent, what formulas we can use for and rules or or rules, that things are or aren’t mutually exclusive or independent. All fine, but then the problem comes, we get to a real question, how do we apply these rules? So this is a video where I’ll start to talk about applying these rules.
First of all, let’s just review briefly again. What can these two terms mean? Mutually exclusive means there is absolutely no way, that these two things can appear at the same time. There is a 0% probability of them showing up side by side. These are often used for things from the same category.
For example, if I pull a single card from a deck. Pulling a heart and pulling a spade are mutually exclusive. Because if I pull only one card, obviously it can only be one suit. It can’t be two suits at once. Similarly, if I roll a single die, the faces of the die are mutually exclusive. When I roll a die once, I can’t possibly get more than one face showing.
So, those are examples of mutually exclusive events. Independent means that finding out the probability of one, tells you absolutely nothing about the probability other. In other words, the result of one outcome has absolutely no influence whatsoever on the results of another. Now, a few considerations about these.
First of all, the conditions of mutually exclusive and independent are not common with people situations. They’re more common with dice, coins, cards, etcetera. What do I mean by that? Well, if you think of the categories into which we put people, we can categorize people by their religion, by their political party, by their race, by their gender, etc.
These are big messy categories. We can’t really say that any one category is mutually exclusive with any other. We can’t say that any one category is independent of any other. Categorizing people turns out to be a very messy business, cuz there are always those exceptions as there are people that are odd combinations of things.
They, they’re, they’re this political party but they happen to be that religion. That sort of thing. Just, human beings are just very quirky and there are always exceptions to patterns, and therefore these nice, neat distinctions are mutually exclusive or independent, they’re almost never true when we’re talking about classifying people.
They work much better when we’re talking about very simple inanimate objects. So again, dice, coins, cards, these are the types of scenarios where we’re actually going to be using the ideas of mutually exclusive or independent. Another big idea just to keep in mind. Whenever you see the words, without replacement. Processes that involve those words, without replacement, automatically are never independent.
Well what I’m, what do I mean by that? Well, suppose we’re choosing cards from a deck without replacement. Well, when I pick the first card, alright, then I can forget the probability that that card is a certain suit or something like that, but then when I go to pick the second card, now instead of 52 cards, there are only 51 cards.
Because one has been picked already and it hasn’t been replaced. So now picking a card out of a deck of 51, that is going to be a different probability then picking a card out of a deck of 52. And in particular, what I pick on the first deck. For example, I won another probability of picking hearts. Like pick a heart of the first draw.
Well, that is going to change the probability of whether I pick a heart on the second draw. And so without replacement, the different draws are going to always have influence on one another and therefore they are never gonna be independent. So that’s just a handy trick to remember, when you see the words without replacement, you’re not going to be using any of the independence rules.
Okay, let’s look at a hard problem. In a certain game, in phase one, you flip one coin as many as three times. If you flip three tails, you lose. You automatically lose. As soon as you get your first head in phase one, you advance to phase two. In phase two, you roll a six-sided die once.
If you roll a six, you win. For any other roll, you lose. What is the probability of winning? Now, of course, this, if this were an actual test question, we’d have to have answer choices, we’re just going to treat this as a calculation here. You may want to pause the video here and extempt word this problem on your own just to get a sense of it.
So this is my abbreviation scheme for the problem. In phase one, we’re gonna have up to three coin tosses. If I toss tail, tail, tail, I lose. As soon as I get the first head, either on the first toss or the second toss or the third toss, then I advance to phase two. Phase two just consists of a single role of one die, if I roll the 6 I win, if I roll everything else I lose.
So first of all, notice that we’re working with coins and dice. Again, very simple things. Whatever I do with the coins is gonna have absolutely no effect on what happens with the die. The die roll clearly is gonna be independent of the coins, and what this means is that phase one is independent of phase two.
So, probability of winning, this is the probability of phase 1, what we get is the advance to phase 2, that’s how I’ll abbreviate it. And in phase two, we roll a six. Okay, that is the scenario that will result in winning and because the two phases are independent, we can multiply these. So, phase one results in advancing to phase two times.
The result in phase two is roll of a 6. Well, one of those factors is very easy to figure out. When you roll one die, what’s the probability of getting a 6? Well that’s a probability of one-sixth. So that means if we could figure out the first one, the probability that phase 1 results in advancing to phase 2.
We would just multiply this by 1/6. So in a way, we’ve half way done with the problem. Now let’s go and look at phase one in a little more detail. Let’s think about the outcomes here. If we flip a head, then I advance. If I flip tail and then head, then I advance.
If I flip tail, tail, head, then I advance. If I flip tail, tail, tail then that’s no good, then I’m out. And I want to figure out the probability of that net. If a direct calculation that would be a little bit tricky. Notice that those three cases would have three different probabilities because they involve a different number of tosses in each case.
But notice that of the four possibilities, three of them are the thing that I want and one of them is not the thing that I want. This is an ideal scenario for using the compliment rule. So the probability of advancing on phase one is one minus the probability of losing. Of course, the probability of losing is the probability of flipping three tails in a row.
Well what’s the probability of flipping three tails in a row? Well tail and tail, and tail, that would be 1/2 times 1/2 times 1/2; which is 1/8th. So the probability of advancing would be 1 minus 1/8th, which is 7/8th. That is the probability that in phase one we wind up advancing to phase two. So that’s our first factor here so we get 7/8 times 1/6, and this turns out to be 7/48, and this is actually the probability of winning the game.
Now this problem here is probably a notch or two harder than anything you’re gonna see on the test. But it demonstrates the basic principles, dividing the problem into stages. And at each point, analyzing are things independent. Are things disjoint. And can we use the complement rule to simplify.
All of these are strategies that will help you analyze probability questions.
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