7.3 Independent events (cloned)
That definition is easy to say, but difficult to really internalize.
When we were talking about the chances of rain on Tuesday and Wednesday in the last unit, we were assuming that the weather on Tuesday doesn’t affect the weather on Wednesday: they are independent of each other. hat’s good, because it allows us to use the “and means times” rule, which only applies to independent events.
But sometimes dependencies aren’t as easy to see as you might assume. For example, the chance of rain Friday absolutely does affect the chance of having it not rain on Friday, but you wouldn’t always even think to even consider that relationship. (Note that there were not two different days in that sentence, perhaps contrary to your expectations.)
So what do you do if the situation doesn’t let you use a rule? Or, worse, if you aren’t really sure whether the rules apply?
Believe it or not, the answer is that you change the way you are looking at the situation so that the rules apply again.
Let’s look at an example of using these various rules wrong, so that you’ll be more able to avoid these mistakes. Along the way, we’ll learn how to change our point of view to allow us to use the rules when we need to.
If there’s a 1/3 chance that it will rain on Tuesday, and a 1/3 chance that it will rain on Wednesday, then what is the chance that it will rain on Tuesday or Wednesday (or both)?
Wrong solution: “or means plus,” so the chance of rain on Tuesday or Wednesday is 1/3 + 1/3 = 2/3. This is wrong; the events are independent.
Second wrong solution: or means plus, so the chance of rain on Tuesday or Wednesday or both is 1/3 + 1/3 + 1/9 = 7/9. (The 1/9 was the chance that it rains on both days.) Still wrong; although the chance of rain on both days is not independent of the rain on either day (good!) it’s still a problem that rain Tuesday is independent of rain Wednesday.
Better solution: Either it rains on Tuesday or not, **and **either it rains on Wednesday or not. So there are four different (independent) events that could happen:
Within each row, the two events are independent, so we can use the “and means times” rule within each row. But each row affects the others, since if row #2 is true, the other rows cannot possibly be true. So we can use the “or means plus” between the rows.
We calculate the chance of each row:
Notice that only one of the numbered events can happen, and one of them must happen. (We say one and only one will happen.) So it is guaranteed to be either #1, #2, #3, or #4. And sure enough, the chances of the four events add up to 100%: 1/9 + 2/9 + 2/9 + 4/9 = 9/9 = 100%.
So this all works correctly.
The chance that it will rain on Tuesday or Wednesday (or both) is 2/9 + 2/9 + 1/9 = 5/9.
Like I mentioned earlier, there aren’t any quizzes for this unit. But if this section reminds you of probability question that you had trouble with in the past, go find it and try to solve it again. Use the ideas above, and plenty of paper. Let’s see what happens.