I’d like to share the stats lessons Sondy from Sonderbooks offered in response to my previous post regarding SLJ Battle Stats 101. They’re too good and geeky to be buried in the comments section. First, she caught my terminology snafu: intersection vs union.
You’re actually not finding the union of the events at the end, but the intersection. In other words, you’re finding the probability that you guess all matches right up to the semi-finals AND the probability you guess the last match correctly. They’re independent, so you multiply, as you did.
So to make sure I really understood (midterm in a week!), I made a diagram:
Sondy then goes on to tackle an interesting problem: the probability that you guess the final winner correctly by any means possible. (Hint: since there’s more than one way to guess the winner, you take the union, or sum, of all the possible winning outcomes.) The answer is surprisingly intuitive. Here’s how she did it:
First, the traditional way of winning. What is the probability that you randomly guess the ultimately winning book to win at each level? In other words, that it wins the first round, second round, semifinals, and finals? That would be (1/2)x(1/2)x(1/2)x(1/3) = 1/24
However, there’s also the nontraditional path, and that way it could have lost in the first round OR the second round OR the third round and then come back from the dead.
The probability you guess the ultimately winning book (let’s call it W) to lose in the first round is 1/2.
The probability you guess W to lose in the 2nd round is the probability it wins in the first round and loses in the 2nd round, (1/2)x(1/2) = 1/4.
The probability you guess W to lose in the 3rd round is the probability it wins in the first and second rounds and loses in the 3rd round, (1/2)x(1/2)x(1/2) = 1/8.
Thus, the probability that you guessed W to lose in the 1st OR 2nd OR 3rd round is 1/2 + 1/4 + 1/8 = 7/8.
The two books that advanced to the finals are NOT eligible, so the probability W is picked [for the un-dead] is only 1/14, not 1/16.
That gives us (7/8) x (1/14) = 7/112
Then the probability it wins the final round that way is (7/112) x (1/3) = 7/336
So the probability should be (1/24) [probability of winning at every level] + (7/336) [probability of losing in round 1, 2, or 3, and then winning when it counts] = (14/336) + (7/336) = 21/336 = 1/16!
What do you know? We could have just said, right at the beginning, that there’s a 1/16 chance of picking the final book correctly.