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The “At Least” Scenario
In this video we’re gonna talk about an incredible time saving shortcut. So first of all we need a slightly broader view. As a general rule, we’ve talked about the Complement Rule already. This Complement Rule can sometimes be used as a shortcut in some problems. In some of the later videos we’ll talk about this more broadly. Using the Complement Rule as a shortcut.
And I’ll say right now, ordinarily, the hard thing about using it as a shortcut is recognizing that one could do this. We read the problem and then we have to have the insight, oh wait a second, maybe it would be easier instead of calculating directly what they’re asking, to calculate the probability that it doesn’t happen, and then subtract that from one.
So once you recognize that,. Then, the Complement Rule will save you time, but you have to have that recognition. Well, here, I’m gonna show you one case where that recognition will not be a problem, it’ll be an automatic clue, and 100% of the time you know, you can use the Complement Rule as a shortcut, so that’s what makes this such an efficient and powerful shortcut.
Here’s the big idea. When you see the words “at least” in a probability question, use the Complement Rule as a shortcut. Now in particular, whenever the words at least appear on the test, it is almost always part of the phrase, at least one. This is a huge advantage for us.
Now why would this be? Let’s think this through. Suppose there are six trials of something. We’re rolling six dice, we’re flipping six coins, something like that. And we want to know the probability of at least one success. Think about the number of successes.
Suppose I’m flipping six coins, how many heads could I get if I flip six coins? Well, theoretically I could get anything from zero heads, if it were all tails, up to six heads, all heads. Now of course, these are not all equally likely but all of them are possible. So, what would be the set, at least one, so at least one would be one or more. All of these are at least one, we could say this box here, that’s the set at least one.
Well what’s the complement of that set? What’s the part of the set, the whole set not included in that box, well of course the complement of that is just zero. And that is always the only case that is not included in the phrase at least one. So this is a huge idea. The complement of at least one is none.
And therefore we can say the probability of at least one success is the probability of zero successes. So in ev, instead of having to figure out a bunch of different probabilities, which is what we’d have to do if we figured it out directly. We can just figure out one probability, often a very, very easy probability to figure out.
The probability of something happening, no times as all. And figure that out and subtract it from one and we’re done. So I’ll show you a couple examples of why this is so powerful. Suppose we have this problem. This is a very hard problem. Suppose we roll one fair six sided die eight times.
What is the probability that we will roll at least one six? So let’s think about this. We have two choices. We can do sort of the forward, straight forward. The completely straightforward method of solving this and as you’ll find this will, that route will be very, very time consuming or we can use the shortcut.
So first, just so you appreciate how much time we will save I’m gonna show you this, this more lengthy straightforward method so, just so you get an idea of what’s involved with this. So, the standard solution. The condition, at least one six, includes eight cases. We could get exactly one six, or we could get more than one six.
So any of the numbers from two through eight, these are also included. These are the number of dice that could show up with a six on it. Again, these are not all equally likely, but it doesn’t matter, all of these are possible cases. Each one of these eight cases, is a binomial calculation. So that means we’d have to do eight different binomial calculations.
So we’re gonna have to do all these calculations. Now, of course, in a problem like this, they’re not gonna give it to you in a nice form where all these things added up. You’re gonna actually have to go through a lengthy calculation, simplify all this stuff. Put it all together as a number, and then choose from the numerical answers.
So this calculation could take you, I don’t know 10 or 15 minutes. This would be a nightmare calculation. If you want to do it go ahead. I’m gonna show you a shortcut that’s much easier. So here’s the shortcut, the efficient solution. The complement of at least one six is zero sixes.
Let’s think about this. Probability of not getting a six on one roll is five over six. The probability of that happening, of getting not a six eight times is five-sixths to the eighth. Until the probability is 1 minus five-sixths to the eighth. Now you may have to calculate this but it may well be that if it’s a smaller number, of course, they, you will calculate it.
But something like this, five-sixths to the eighth, they’re probably not gonna ask you to calculate that, they’re probably just going to write the answer in this form, one minus five-sixths to the eighth. So if you did it the forward way, you’d have to realize that you could simplify that whole mess down to something as simple as this.
But it turns out, the way we did it here this efficient solution. We got there with almost no calculations. In other words this was just a ridiculously easy way to get to an answer here. So, in summary, I will say whenever you see the words at least in a probability problem.
And especially when you see the words “at least one”. Then you know automatically that the shortest and most efficient solution will be used in the Complement Rule, using the fact that the complement of at least one is zero. Now, again in future videos we’ll talk about using the Complement Rule in more complicated scenarios, in, in the other scenarios where it would be a short cut.
There’s gonna be a problem of recognizing it, for this particular case, there should be no problem recognizing it, you’ll see these words and that will be the trigger, automatically you know you can use the Complement Rule as a shortcut.
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