Casing
Casing
If you can break a problem down into several smaller (easier) pieces, you should. Always.
A common way to do that is to notice that a situation can be divided up into a few different cases. For example, you might notice that there are only three different ways a situation can unfold. So, you might consider those three situations – those three cases – separately. Congratulations! You’ve just turned a problem into three easier problems. That’s always better.
That’s because working out multiple simple cases is pretty much guaranteed to be easier than trying to work through a single more-complicated situation.
A “simple” example
If there’s a 60% chance of rain on each of Monday and Tuesday, what’s the chance that it will rain on exactly one of those days?
The key here is to recognize that there are actually four different weather patterns that we could observe over these two days:
- Raining both days
- Raining on Monday
- Raining on Tuesday
- Not raining on either day
These are our four cases. According to the problem, we are interested in knowing the probability that either Case 2 or Case 3 occurs.
But, as suggested earlier in this chapter, even the simple probability problems are generally complicated. This one is no exception, since not only do we need to consider all four of these cases in order to keep everything straight, but we also need to recognize that in order for it to “rain only on Monday” it needs to not only rain on Monday, but also it needs to not rain on Tuesday.
So our cases really look like this:
- Raining both days
- Raining on Monday but not on Tuesday
- Raining on Tuesday but not on Monday
- Not raining on either day
Once we’ve clarified the meanings of our cases, we can get to calculation. (This means a tiny bit of enlightened brute force.)
Case 2: 60% chance of rain on Monday, times the 40% chance of rain on Tuesday, means a 24% chance that Case 2 occurs.
By symmetry, Case 3 also has a 24% chance of occurring (i.e. both Case 2 and Case 3 involve multiplying 60% by 40%, whose answer doesn’t change in between the calculations).
Our answer appears to be the sum of the two cases (i.e. 48%), since we are interested in the chance that either Case 2 or Case 3 occurs.
But just to make sure, let’s check one other thing. We know that if we’ve done this correctly, one of the four cases is guaranteed to occur. That means that the four cases should add up to 100%. So if we are right about Cases 2 and 3, then Cases 1 and 4 should add up to 52%.
Case 1: Raining Monday and raining Tuesday means 60% times 60%, which is 36%. Case 4: Not raining Monday and not raining Tuesday means 40% times 40%, which is 16%.
Sure enough, 36% plus 16% is 52%, so our final (correct) answer to the original question is indeed 48%.
What to do
Like I mentioned earlier, there aren’t any quizzes for this unit. And I hate to say it, but this really is one of those areas where you just need to practice.