Notes for March 11

Confidence of victory

The margin of error is the industry standard, but the people who use these numbers, most especially the news media, really don't understand them very well. Here is a different method to produce a more useful piece of mathematical information which this author has developed, called the confidence of victory method. Here is a recent poll from Public Policy Polling (PPP) on the North Carolina senate race.

n = 884
Hagan 45%
Tillis 43%


These numbers add up to 88%, which means about 12% are either undecided or prefer another candidate. What we do is effectively ignore the respondents who are either voting for third party candidates, are preferring none of the candidates or are still telling pollsters they are undecided. We now figure out how many people in the poll said they prefer Hagan and how many prefer Tillis by multiplying the percents by the size of the poll.

f(Hagan) = 884 * .45 = 397.8
f(Tillis) = 884 * .43 = 380.12

new n = 397.8+380.12 = 777.92

p-hat(Hagan) = 397.8/777.92 ~ 51.1%
p-hat(Tillis) = 380.12/777.92 ~ 48.9%

sp-hat = sqrt(.511*.489/777.92) ~ 1.79%

z(Hagan) = (51.1 - 50)%/1.79% ~ .61

This says Hagan's percentage is about 0.61 standard deviations above 50%. The percentage he will get in the actual election may be higher or lower than what we see here. We assume there's about a half a chance he will do better than the final opinion poll, and a half a chance he will do worse. What the public actually cares about is whether he wins or loses. What the confidence of victory method does is find the percentage that corresponds to the z-score. That number is the confidence level we have that the true percentages from the population polled will show that the leader in the poll will be the winner of the election. In this example, z=0.61 corresponds to .7291 on our positive z-score table. Because the confidence of victory method is sensitive to small changes, we should round to the nearest percent and use this sentence to describe the results.

If the election were held when the poll was taken, we are 73% confident that Hagan will hold on to the lead shown in the poll and win the election in North Carolina.

In 2008, the final polls in the 50 states and Washington D.C. had two states too close to call, Missouri and Indiana. Both elections were very close, called late in the evening, Missouri for McCain and Indiana for Obama. In the other 49 contests where confidence of victory claimed an advantage for one side or the other, 48 contests were won by the person leading in the most recent poll, which is to say the confidence of victory method was vindicated about 98% of the time. The only state where the confidence of victory method did not get the right result was North Carolina. McCain had a 60% confidence of victory in North Carolina, but Obama actually won the state.

In 2012, confidence of victory did much better. There was only one toss-up state, Florida, a state that did not give their results for four days. All the rest of the states were called correctly in the electoral college vote, as were all the U.S. Senate races.




Probabilities and payoffs

The probability of winning a game may be known or unknown, but that is different from the payoffs of the game, sometimes called the odds. There are two methods, which we will call the classic parimutuel  and the modern parimutuel.

Classic parimutuel: The statement has two numbers, sometimes written with a colon between them, such as 7:1, or sometimes a dash such as 7-1 or sometimes the word "to", like 7 to 1. In all these cases, the first number represents a ratio of the profit a player will get if the bet goes well and the second number is the amount at risk should the gamble fail. In any gamble, there is another person (or maybe a casino or a bookmaker) on the other side of the bet, and the profit and risk are reversed.

Example: 7:1 odds (first person's perspective)
If you get 7-1 odds and risk $10, the opponent must put up $70. The $80 is put someplace for safe keeping until the result is posted. The winner will get all $80. For the person getting 7:1 odds, there would be a $70 profit in case of a win, but the other side, who looks at the bet as a 1:7 proposition, there would only be a $10 profit in the case of a win.

Classic parimutuel is always given in numbers written in lowest terms, so 70:10 would be written as 7:1, or 25:10 would reduced to 5:2. Some like to have the low number be 1, so 5:2 will sometimes be written as 2.5:1.

Modern parimutuel: In modern parimutuel, only one number is given either a positive like +150 or a negative like -170. We need a second number, and in this system the second number is always 100.

Positive example: If the number is positive like +150, this number is the profit and the risk = 100.

Negative example: If the number is negative like -170, the number 170 (the absolute value) is the risk and the profit = 100.

Changing modern to classic: For +150, we would write the classic as 150:100 (profit:risk) and reduce the fraction to lowest terms, so here it would be 3:2.

For -170, the ratio would be 100:170, which would reduce to 10:17 in lowest terms.

Changing classic to modern: We will have two numbers, profit:risk and we are interested in which is smaller. What we will do is multiply both numbers by 100/small, and the number that is largest is the modern parimutuel, with a + sign if profit > risk and a negative sign if profit < risk.

Profit > risk: If the classic odds are 5:4, multiply both numbers by 100/4 = 25 to get 125:100. Because the big number is first, the modern version would be written +125.


Profit < risk: If the classic odds are 5:8, multiply both numbers by 100/5 = 20 to get 100:160. Because the big number is second, the modern version would be written -160.

The expected value of a game

The expected value of a two outcome game (win or lose) can be written as EV= p*(profit+risk)/risk. By dividing by risk, the game's expected value can be thought of as a percentage of money returned to you on average every time you play. Again, this is an average, so the average outcome doesn't have to be achieved. In many games, it never is. Expected value is really about the long run.

Flipping a fair coin, if you call "heads" every time, you should win about 50% of the time, and so EV = .5(1+1)/1 = 100%, so calling heads is a way to make this a fair game with a fair coin.

If you mix up your calls, it's also a fair game.

If you go with "rock" every time in rock/papers/scissors, your opponent will catch on soon enough and go with "paper" every time, and you will lose money in the long run.

Rock/scissors/paper is a fair game only if you can mix up your calls using some random method, or at least a method hard for your opponent to determine. The best method is 1/3 rock, 1/3 scissors and 1/3 paper, and the expected value is 100%, or that you will neither lose nor win, but break even.

Let's go back to roulette. We saw that playing a single number or playing either black or red produce the same expected value for the player.

Single number for player
p = 1/38
profit = 35
risk = 1
EV = 1/38*(35+1)/1 = 36/38 ~= 94.7%

Red (or Black) for player
p = 18/38
profit = 1
risk = 1
EV = 18/38*(1+1)/1 = 36/38 ~= 94.7%

For the casino, the probability of winning is 1 minus the probability for the player. The risk and profit numbers are switched from the players' values.

Single number for casino
p = 37/38
profit = 1
risk = 35
EV = 37/38*(35+1)/35 = (36*37)/(35*38) ~= 100.15%

Red (or Black) for player
p = 20/38
profit = 1
risk = 1
EV = 20/38*(1+1)/1 = 40/38 ~= 105.3%

When profit = risk, which is the same as saying the classic parimutuel odds are 1:1 or the modern parimutuel odds are +100 (which is the same as -100, though rarely written that way), the average of the two expected values will be 100%. You expect to lose about 5.3 cents every game and the casino expects to win about 5.3 cents.

When profit does not equal risk, we get different percentage advantages and disadvantages. Because the casino must risk 35 bets and the player only 1 bet when the player picks a single number at roulette, the expected value for the casino is still positive, but relatively small at .15 of a cent per game. In reality, the casino rarely spins the wheel with only one bettor playing, so the casino is not truly risking only its own money on a single spin, but can use the losses of some players to help offset any possible winner. Even if that weren't the case, a game with an expected value greater than 100% means a winner in the long run, and that is the business model the casino operate on, and very successfully as anyone can see.