The Law of Averages

What does the law of average say? We know that, from the definition of probability, in the long run the frequency of some event will be around the probability of the event. For example, a coin lands heads with probability 50%. So after many tosses, the number of heads should approximately equal the number of tails. Is the law of average all about this?

Question Suppose we are tossing a coin. If we get a lot of heads, then tails start coming up. Or if we get too many tails, the chance for heads goes up. In the long run, the number of heads and the number of tails even out. True or false? Why? Answer: False.

Explanation With a fair coin the chance for heads stays at 50%, no matter what happens. We learned this from independence. So whether there are two heads in a row or twenty, the chance of getting a head next time is still 50%. Moreover, since the probability keep fixed, if we get a lot heads at the beginning, the number of heads and the number of tails may not even out at the end. Instead, the chance error will go up in absolute terms. Here chance error means the amount off the expected value. For the tossing coin example, number of heads = half the number of tosses + chance error.

Explanation This will not contradict to the probability. Because in the long run the chance error only gets bigger in absolute terms, however, compared to the number of tosses, it gets smaller. For example, in an experiment a coin was tossed for 30, 500, 5000, times, the chance error was 2, 5, 33, 67. This shows the chance error gets bigger. But the percentage gets smaller 6.67%, 1.00%, 0.66%, 0.67%. A natural question will be: how big the chance error is likely to be? For example, with 100 tosses, the chance error is likely to be around 5 in size. With 10,000 tosses, the chance error is likely to be around 50 in size. Multiplying the number of tosses by 100 only multiplies the likely size of the chance error by √100 = 10. (We will discuss this later.)

The table for the experiment

The graphs for the experiment

Summary of the law of averages As the number of tosses goes up, the difference between the number of heads and the expected number (half the number of tosses) gets bigger. That is the chance error gets bigger. But the difference between the percentage of heads and the probability (50%) gets smaller. That is the ratio between the chance error and the total number of tosses gets smaller. The likely size of the chance error will get bigger, but the rate will not be as fast as that of the total number of tosses.

Chance processes In the previous example, we met with the problem of chance variability: when a coin is tossed a large number of times, the actual number of heads is likely differ from the expected number. Until now, we still don’t know how to calculate the likely size of the difference (chance error). These problems are about chance processes: chance comes in with each toss of the coin. If we repeat the experiment, the tosses turn out differently, and so does the number of heads.

Chance processes In order to study all kinds of chance processes (e.g. tossing a coin, playing the roulette, drawing a random sample from a large population), we need to generalize a standard model to analyze how the numbers influenced by chance: The box model----drawing numbers at random from a box. (It is easier for us to analyze mathematically.) The idea is that: Find an analogy between the chance process and the box model. Connect the variability with the chance variability in the sum of the numbers drawn from the box. We will see how to use this idea from the examples later.

The sum of draws Let us look at our standard model process: There is a box of tickets. Each ticket has a number written on it. Some tickets are drawn at random from the box, and the numbers on these tickets are added up.

Example Suppose we have a box with tickets: 1, 2, 3, 4, 5, 6. We draw twice at random with replacement from this box. We then add up the two numbers. For example, we may draw 3 at first, then 5 the second. So the sum is 8. Or the first draw may be 3, and the second may be 3 again. So the sum is 6. There are many other possibilities. The sum is subject to chance variability. Notice that this process is the box model for rolling a pair of dice, then add up the two numbers.

Another example Now we make 25 draws from the same box: 1, 2, 3, 4, 5, 6. Then study the sum of the numbers drawn from the box. We programmed the computer to make the draws: The sum is 88. If we had the computer repeat the whole process ten times, the sum would have been different: The chance variability is easy to see. The values range from 78 to 95.

Another example In principle, the sum could have been as small as 25 x 1 = 25, or as large as 25 x 6 = 150. But in fact, the ten observed values are all between 75 and 100. Would this keep up with more repetitions? What is the chance that the sum turns out to be between 75 and 100? We will study this kind of problem later in the course. This box model tells us how to use the simplified model to analyze the process we want to know about.

Making a box model The sum of the draws from the box turns out to be the key ingredient for many statistical procedures, so keep your eye on the sum. Before we make a box model, try to ask the following question first: What numbers go into the box? How many of each kind? How many draws? The purpose of a box model is to analyze chance variability, which can be seen in its starkest form at any gambling casino. So let us focus on the box models for roulette.

The Nevada roulette A Nevada roulette wheel has 38 pockets: One is numbered 0, another is numbered 00, and the rest are numbered from 1 through 36. The croupier spins the wheel, and throws a ball onto the wheel. The ball is equally likely to land in any one of the 38 pockets. Before it lands, bets can be placed on the table.

A Nevada roulette table Let us focus on the red or black first. Red or black: except for 0 and 00, which are colored green, the numbers on the roulette wheel alternate red and black. If you bet a dollar on red, say, and a red number comes up, you get the dollar back together with another dollar in winnings. If a black or green number comes up, the croupier smiles and rakes in your dollar. The case that you bet a dollar on black is similar.

Convert to a box model If you bet a dollar on red, and the croupier spins the wheel. It may seem hard to figure your chances. But a box model will help. The numbers go into the box: You will either win a dollar or lose a dollar. So the tickets must be either +1 or -1. How many of each kind: We know 0 and 00 are green, and numbers from 1 through 36 alternate red and black. So you win if one of the 18 red numbers comes up, and lose otherwise. Therefore, your winning chance is only 18 in 38, and the chance of losing is 20 in 38. There are 18 “+1” tickets, and 20 “-1” tickets in the box.

Convert to a box model As far as the chances are concerned, betting a dollar on red is just like drawing a ticket at random from the box. The advantage of the box model is that all the irrelevant details----the wheel, the table, and the croupier’s smile----have been stripped away. The cruel reality is that you only have 18 tickets and they have 20.

The net gain Suppose you play roulette ten times, betting a dollar on red each time. You will end up ahead or behind by some amount. This amount is called your net gain. The net gain is positive if you come out ahead, negative if you come out behind.

Box model for net gain On each play, you win or lose some amount. This is modeled by drawing a ticket from a box. In ten play, it is just like ten draws from the box, made at random with replacement. So the net gain----the total amount won or lost----is just the sum of these ten win-lose numbers. So the box model will be the sum of ten draws made at random with replacement from the box with 18 “+1” tickets and 20 “-1” tickets.

Example Suppose, for instance, that the ten plays came out this way: R R R B G R R B B R (R for red, B for black, and G for green.) We have the following table for the net gain: When you get a red, the win-lose number is +1, and the net gain goes up by 1. When you get a black or a green, the win-lose number is -1, and the net gain goes down by 1. The net gain is just the sum of the win-lose numbers, which is just like the sum of draws. This example has a happy ending: you come out ahead $2. We will study later about the chance if you keep on playing.

Another example When you bet a dollar on a single number at Nevada roulette, and that number comes up, you get the $1 back together with winnings of $35. If any other number comes up, you lose the dollar. Gamblers say that a single number pays 35 to 1. Suppose you play roulette 100 times, betting a dollar on the number 17 each time. What is the box model?

Solution The first question is what numbers go into the box: In one play, you will win $35 dollars only when the ball drops into the pocket 17, otherwise you lose $1. So the tickets will be “+35” and “-1”. Comment: The tickets in the box show the various amounts that can be won or lost on a single play.

Solution The second question is how many tickets of each kind: In one play, you win only when then number 17 comes up. It is only 1 chance in 38 of winning. For the other 37 chances in 38, you lose. So there is 1 ticket for “+35” and 37 tickets for “-1”. Comment: The chance of drawing any particular number from the box must equal the chance of winning that amount on a single play. (“Winning” a negative amount is the mathematical equivalent of what most people call losing.)

Solution The third question is how many draws: Since you are playing 100 times, the number of draws has to be the same Tickets must be replaced after each draw, so as not to change the odds. That is we draw ticket with replacement. Comment: The number of draws equals the number of plays.

Solution So the net gain in 100 plays is like the box model: Sum of 100 draws made at random with replacement from the box with 1 ticket for “+35” and 37 tickets for “-1”.

Summary There is chance error in the number of heads: Number of heads = expected value + chance error. The error is likely to be large in absolute terms, but small relative to the number of tosses. This is the law of averages. In percentage terms, the percentage of heads is likely to be close to 50%, although it is not likely to be exactly equal to 50%. The law of averages does not work by changing the chances. For example, after a run of heads in coin tossing, a head is still just as likely as a tail.

Summary A complicated chance process for generating a number can often be modeled by drawing from a box. The sum of the draws is a key ingredient. The basic questions to ask when making a box model: What numbers go into the box? How many of each kind? How many draws?

Summary For gambling problems in which the same bet is made several times, a box model can be set up as follows: The tickets in the box show the amounts that can be won(+) or lost(-) on each play. The chance of drawing any particular value from the box equals the chance of winning that amount on a single play. The number of draws equals the number of plays. The net gain is like the sum of the draws from the box.