Warriors 3-1 Comeback

I happened to be in the bay area this weekend. The bay area the home of the Golden State Warriors. A few days ago it wasn't looking good for them. Oklahoma City was ahead three games to one in their best of seven series. So to win, Golden State would have to win three games in a row.

There were a million sports radio shows that had their predictions of the likelihood of Golden State winning three in a row. Most all of them decided that it was not likely. But somehow they did. I like to think my presence in the bay area had a little to do with those wins. What were the chances anyway? Virtually all of the radio experts were making basketball predictions. However, it is also a good math application as well.

Theoretically I would think that 0.5 x 0.5 x 0.5 = 0.125 might be it. So the team with the lead has a 87.5% chance of winning the series. Historically, counting Golden State's win, the team that was ahead has now closed it out 95.7% of the time. It's close, but quite a bit ahead of our theoretical figure. But that does make some sense. We were assuming the chance of a single win is 50%. But if your team is ahead three games to one, you probably (but not necessarily) have the better team. Maybe that team against the same opponent would win 60% of the time. Then 0.4 x 0.4 x 0.4 = 0.064. That means the leading team would have a 93.6% chance of winning the entire series. Now we're getting there.

In fact, what winning percentage would correspond with the 95.7% figure? If the chance of winning three straight is .043 (1 - .957), then the chance of winning one game is the cube root of .043 or .350. That would mean that the team ahead in the series would normally beat the other team in a single game 65% of the time. That seems fairly reasonable.

I found a bunch of cool historical stats like this on the following website: http://www.whowins.com/tables/up31.html

I'm sure there are others like this with different scenarios. This one states the chances of winning if a team is up 3-1 in the playoffs in various settings.

• In the NBA it is now 95.7%. 
• In major league baseball it is only 85.2% (slightly less than the theoretical percent).
• In the NHL it is 90.1%
• In all three sports combined it is 91.7%

This might be of a bit more interest to students before the series is over, but regardless, I think its a pretty cool application.

More Batting Average

Continuing from last week's math application, we know that a player's batting average is found by dividing his hits by his times up to bat.

There was just a news article how Jon Lester, a fine pitcher, but poor hitter for the Cubs has started the season at 0 for 59. In batting average talk this is not called "zero", but .000. This is a record to start things off this badly.

Of those getting up to the plate over three hundred times, the record is Dean Chance at .066. Another poor batter was Fred Gladding who went one for 63.

Students might have heard that Ted Williams was the last to "hit four hundred". This means during that particular season (1941) his batting average was over .400. He was 185 for 456 which is .406.

Students will have heard of "batting a thousand". This means a hit every time at bat, which is mathematically actually batting "one", but it is written as 1.000 and in baseball it is called batting one thousand.

Here is batting average problem with a system of equations. You start rooting for a player that mid season has a batting average of .312. From that point you keep track and he goes 40 for 100 and ends the season with a batting average of .341. How many hits and at-bats did he have when he was hitting .312?

x/y = .312   and    (x+40)/(y+100) = .341.

Even rounding off to thousandths place there will be some round off error and final answers may need to be adjusted a bit. Obviously, at-bats and hits will have to be whole numbers.

Batting Averages

Batting averages are a nice math application for those starting out learning algebra. Some students will think this is a pointless application, while others will think, "Hey, this stuff is useful after all."

A player's batting average is found by dividing the number of hits by the times at bat. It is rounded to the nearest thousandth place. It is never pronounced as it should be. While 0.250 should be pronounced "two hundred fifty thousandths", baseball people pronounce it "two-fifty". A person with a batting average of 0.267 is said to be hitting "two sixty seven". And so on.

If two of the pieces are known, the third can be found with a little algebra. As of this moment, these numbers are current numbers for these players:

David Ortiz has 41 hits in 128 at-bats. What is his batting average? (41/128 = x; x = .320)

Zach Cozart has 113 at-bats and is hitting .319. How many hits does he have? (x/113 = .319; x = 36)

Eric Hosmer is batting .336 with 47 hits. How many times at bats does he have? (47/x = .336; x = 140)

Also of interest might be the fact that sometimes thousandths place isn't enough accuracy. In 1949, George Kell won the batting title over Ted Williams even though they were both listed as batting .343. George was 179 for 522 (0.3429) and Ted was 194 for 566 (.3428).

Pretty much the same thing happened in 1970, with Alex Johnson (.3289) defeating Carl Yastrzemski (.3286) by a whisker.

Marathon Percentages

I came across this in Runner's World magazine (Page 88, May, 2016). It seemed like a good application for a math class. It had the percentage of women in the field of the Boston Marathon. It seemed like it was something close to a linear relationship. Here is the data:

Year          # Women          # Men          % of Women

1966          1                      415               0.3%
1972          8                      1,210            0.7%
1980          237                  3,428            6.5%
1990          1,434               6,516            18.0%
2000          5,469               10,199          34.9%
2010          9,560               13,161          42.1%
2015          12,018             14,580          45.2%

I tried it out on an on-line calculator. First it is years vs. percentage of women. I used the point (0,0.3), (6,0.7), (14,6.5), etc.

The best fit was a linear equation of y = 1.0232x-3.8943. It had a correlation of 0.9866, so pretty good. I graphed it and perhaps some kind of logistic growth model perhaps some kind of logistics growth model might be a little better. The rate of increase of the percentages is starting to fall off a little.

I wasn't going to do this, but for fun I graphed years vs. number of women. It looked fairly parabolic. After plugging them in - (0,1,), (6,8), etc. I got the best fit to be:

y = 0.3888x^2.5833 with a correlation of 0.9751.

So, some interesting applications could be looked at from Algebra I through Advanced Math.

Who Wants to Be a Millionaire II

Last week's application dealt with a mathematical situation involving the winnings board on the game show Who Wants to Be a Millionaire? The question was whether there is a best fitting curve that fits the numbers fairly well.

We just kind of left it right there. Again, I apologize for doing this in Euros or what ever those numbers are. I guess that just gives another opportunity to do the project in dollars.

I used a website http://www.had2know.com/academics/regression-calculator-statistics-best-fit.html. It allowed you to input ordered pairs and then would give you four answers: Linear, Exponential, Power, and Logarithmic. The are other websites that have even more options.A teacher could get a lot of mileage out of this. Students could take guesses at which of the four would be best and which would be worst, and what the winning equation might look like. Enough suspense. The equations the website came up with are as follows:

Linear: y = 42877.1492x-209477.1429
                    Correlation:  0.6949

Exponential:  y = 41.5145(1.9468^x)
                    Correlation:  0.9992                  

Power:   y = 11.9805(x^3.5337)
                     Correlation:  0.9267                  

Logarithmic: y = -202534+180689.7131lnx
                      Correlation:  0.512

The winner? Judging by the correlations, the exponential equation, although the power equation wasn't bad. The others, fairly bad.

Students could graph the original 15 ordered pairs along with these four equations for some conformation.