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In order to talk about the probability AND rule, we need to talk about the idea of Independent Events. So what does this mean? In probability two events are independent if they have absolutely no effect on each other. The outcome of one has absolutely no bearing or influence on the outcome of the other.
So for example in very simple systems if we roll two dice, well then the number of dots on one die is absolutely independent of the number of dots of the other die. Whatever happens on one die it’s gonna have no influence whatsoever on what happens on the other die. Successive flips of a coin, or two or more coins flipped together. In other words if I flipped a nickel and then flipped the nickel again.
Whatever happens on the first flip is gonna have absolutely no effect on the second flip. Of course, a coin has no memory, no resentment, nothing like that. So it’s gonna have no way of being influenced by the previous flips. And similarly if I flip several coins at the same time, whatever the outcome is on any one coin has no influence at all on the other coins.
So the idea of independent is often applied in these very simple systems. Now we can also talk about it in real world scenarios. But in real world scenarios it tends to be somewhat bizarre combinations. So for example, the number of emails I receive in a day and the Dow Jones Industrial Average on that day. These two things are entirely unrelated, in other words if you know one, if you have information about one, you have zero information about the other.
Similarly, the number or runs the New York Mets score in a particular day and the number of fish caught in the Osaka region during that day. Again, two things that are 100% unrelated. And so, yes, these are independent. But notice here, when we’re talking about real world events that are independent, typically we’re talking about things that are so obviously unrelated that is not even particularly interesting to talk about them both at the same time.
And so the idea of independent is somewhat less common when we’re talking about meaningful real world events. When we talk about playing cards, one where we can talk about independent is where if I draw one card from a full deck, the suit is independent of a card number. In other words, if I tell you that I picked a card from a full deck and I picked say a heart.
Knowing that I picked the heart that would give you absolutely no information about what the number of the card was. Whether it was a numbered card, a face card, anything like that. You’d have no information about that if all I told you was the suite, so in that sense the suite of the card, and the number of the card are independent. Now, if I draw more than one card, then we have to talk about this issue of with replacement or without replacement, these are words that you will see in probability scenarios, so it’s very important to understand these terms.
With replacement means; that whenever a selection is draw, we put it back into the deck and make the next choice from a full and newly shuffled deck. So each and every choice comes from a full and shuffled deck. So this is what would be to pick cards with replacement, I would pick one card; look at it, put it back into the deck, shuffle the deck, and so that the second card I pick is also coming from a full shuffled deck.
That is what it means to pick with replacement. If I’m picking out of a hat, means I pick one ball from the hat, record it, put it back into the hat so that the second choice would be from the same hat full of balls. Without replacement, well this is what actually happens in real life card games. We pick a card and put it aside.
For example, if you’re dealing cards to people’s hands in, let’s say a poker game or something like that, well that is selecting cards without replacement. That is, once you take the card out, it is no longer in the selection pool. That particular card can no longer be selected anymore. Because it is out of the deck and every new card gets picked from a diminished deck.
So when cards or anything are picked with replacement, all choices are made from the same pool, so each new choice is independent of previous choices. So with replacement means that the successive choices are independent. If cards or anything are picked without replacement, that means each new choice is made under different conditions. Each choice changes the probability for all successive choices, so the choices are not independent.
With card, for example, each time I pick a card without replacement. That means that the next card picked will be picked from a different deck. First I’ll be picking from 52 cards, then I’ll be picking from 51 cards, then from 50 cards, then from 49 cards, and so forth. Each new choice is from a different number of cards in the deck. So if A and B are independent, and this includes selections with replacement.
If A and B are independent events then we can use the very simplified AND rule and simply means multiply. So the probability of A and B equals the probability of A times the probability of B. This is the very simple rule. So, summary first of all it’s very important to determine independence, so really the crucial question is does A, the way A turns out have any affect whatsoever on the way B turns out and if the two events have no effect on each other.
In other words, if any outcome of one can go equally well with any outcome of another, then the events are independent. If the events are independent, then we can use the simplified AND rule: the probability of A, and the probability, the probability of A and B equals the probability of A times the probability of B.
If events are not independent, then we have to use a much more complicated probability rule, which we’ll talk about in upcoming videos.
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