مثالی از قانون AND

سرفصل: بخش ریاضی / سرفصل: احتمالات / درس 7

بخش ریاضی

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Examples of the AND Rule

In the last video we’ve learned about the simplified AND rule. AND just means multiply when we’re dealing with independent events. So, in this video we’re going to look at some examples of this. So, first of all, what is the probability of tossing three coins and getting heads, heads, heads? Well, first of all, notice that the three coins are independent, whatever happens to one coin is gonna have no influence on another coin.

So, we have three independent events, probability is one half for getting heads in one case, one half, one half, and so the three one halves just get multiplied for a probability of one eighth. One eighth is the probability of tossing three heads. Another question.

What is the probability of rolling two six-sided dice and getting snake eyes? That’s one on each die. So snake eyes would look something like this. Well, again, the two die don’t influence each other. So, these are independent events also. The probability of getting a one on one six sided die is one sixth, so one sixth on one, one six on the other, they’re independent so that means and means multiply.

Multiply them we get 1 over 36, the probability of rolling snake eyes when you roll 2 dice. Three cards are selected from a full deck, each time with replacement, what is the probability of selecting three spades in a row? So again. Let’s make sure that we are perfectly clear on this idea of with replacement.

What that means is I shuffle the deck, I pull a card out, I look at it. Then I put that card back in the deck, shuffle it again. So that the second card I pick is being picked from a full shuffled deck. There are 52 cards what I make the second pick. Now I’ll look at that second card, I record it, I put it back into the deck, I shuffle again, so now when I pick the third choice I’m also picking from a full deck of 52 cards, a full shuffled deck, so all three cards are picked from exactly the same conditions.

A full shuffled 52 card deck. That’s what’s going on with replacement. Well, of course, in a modern deck there are four suits. There are hearts, diamonds, clubs and spades and those are evenly divided. So the chance of picking one spade is one quarter. And then, on the second pick it’s also one quarter.

Then on the third pick it’s also one quarter, because we’re selecting with replacement because each time the choice is made from the same initial conditions the events are independent, the choices are independent. And that’s always true with replacement. That means if we get to multiply, we get 1 over 64.

The probability of selecting three spades in a row. Finally, we get this algebraic equation. We don’t even know what the events are. But we’re just told that they’re independent. Sometimes problem will just tell you A and B are independent. Probability of A is 0.6, probability of B is 0.8, what does the probability of A or B mean?

Okay, well this is a bit of a curve-ball. We are talking about an And rule and here we have an Or rule, what’s going on here? Well let’s think about this. This is the general Or rule. And incidentally, if the events are independent that does, that means they can not be mutually exclusive.

If the events are mutually exclusive they are definitely having an effect on one another. Because if one happens it prevents the other one from happening. So that is definitely a very different case from independent. If they’re independent, they’re not mutually exclusive, we need to use the generalized and rule.

So, let’s think about this. We know the probability of A, we know the probability of B. The probability of A and B, well we can get that, because the events are independent, we can get this by multiplying. That would be 0.6 times 0.8, the probability of A and B would be the probability of A times the probability of B, 0.6 times 0.8 is 0.48.

So now we know all 3 terms and now we can say we have 0.6 plus 0.8 minus 0.48. So this is 1.4 minus 0.48 and this winds up being 0.92. Which is the probability of A or B.

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