Fun with statistics (yes, I feel nerdy today)

Walter Ray Williams Jr. is unarguably one of the best bowlers ever, and arguably THE best bowler ever. He has a degree in Physics, and likes statistics as well. I was thinking about the chances our team has of winning our bowling league on Wednesday night, and so I'm going to e-mail this blog to Walter and hope he posts a reply, or responds to my e-mail.

We are currently sitting in 3rd place in our league. Here are the standings. Wednesday is our final night, and a position round which means #1 bowls #2, #3 bowls #4, etc. We bowl 4 games a night, there are 2 points for each game, and 2 points for total pins, so a total of 10 points a night. A quick look at the standings and you'll see that if we win all 10 points, and "Shoe Up!" wins only 2 points (1 game), we'll tie for first and get to have a playoff. If "Shoe Up!" wins more, or "Aggressive Flavor" wins them all, we can't catch them. On the other hand, if we win all 10, noone else can catch us. So I wanted to try and figure out what the odds are that we tie for first, and then win the playoff. Of course keep in mind, when I took a graduate level statistics course at SMSU I got a B. That's the only B I ever got in a math course, and I had plenty of math working on my BS in Electrical Engineering at OC. I attribute it to not being very motivated, but it's still kind of embarrasing. So...

I assume each game to be a statistically independent event. I guess arguments could be made that mental momentum, etc., would change this, but I won't worry about that. I do not take into consideration any teams tying either. For each game, handicap is pretty close to the same for all the teams involved, so that is another variable I have considered insignificant. For the match between #1 and #2, there are 16 possible outcomes for 4 games. If we were to list this for the first place team as W or L for win or loss, the outcomes could be WLLL, WWLL, WWWL, etc. Out of these 16 outcomes, only 4 will work for us, the 4 with 1 win... WLLL, LWLL, LLWL, or LLLW. So the odds of this is 4/16, or 0.25. Given one of these 4 outcomes, we need the odds that second place also wins totals, which I think would also be 1/2 (think of totals like one big game). So the odds that everything goes right for us with the match between first and second is 0.25 * 0.5 = 0.125.

Now the odds that we sweep our match. Pretty simple here, 16 outcomes again, and the only one acceptable is WWWW. So the odds we sweep are 1/16 = 0.0625. Given we win each game, we automatically win totals, so we are done with the odds of winning what we need to. So the odds that both matches turn out the way we want are 0.125 * 0.0625 = 0.007813. Then of course we would have a playoff, and since we don't have any league rules governing this, USBC rule #113b states that the playoff must be the same 4 game format that the normal season used. The chances of us winning the playoff are 1 in 2. This seems intuitive, but can also be derived using the logic I did for the other matches. There is 1/16 way we can win all four games, 4/16 ways to win 3 games, and 6/16 ways to win 2 games. Anything less than 2 we will lose the playoff even if we win total pins. If we win 4 games or 3 games, we win the playoff even if we lose totals. If we win 2 games, we win the playoff only if we win totals, which we again say is a 50% chance. So we can compute the probability of winning the playoff as the sum of weighted probabilities that we win the playoff in any of these three fashions (by winning all 4, winning 3, or winning 2). This gives us the equation (1 * 1/16) + (1 * 4/16) + (0.5 * 6/16) = 0.5. So that is my "proof" that we have a 50% chance to win the playoff.

Now, the overall chance of us winning this league is equal to the chance of both matches going the right way, and us winning the playoff, or 0.007813 * 0.5 = 0.003906. Cool, we have almost half a percent chance to win! Now that I've done this, I'll be even more excited if our team is able to win.