Understanding the Mathematics of Personal Finance

# PROBABILITY AND EXPECTED RETURN

In the chapter on Life Insurance, I presented the idea of the probability of a random event (when you are going to die) and the calculation of an expected value (the average of when many thousands of people just like you are going to die). I’d like to extend these ideas to games of chance.

The expected value of my return on a game of chance is the sum of all the pos­sible things that could happen multiplied by the probability of each of them happen­ing. That sounds worse than it is. Look at the simple example of a coin flip game.

One of us flips a coin. If the coin lands heads up (heads), you give me a dollar; if it lands tails up (tails), I give you a dollar. In terms of the money in my pocket, giving me a dollar is +\$1 and giving you a dollar is a dollar leaving my pocket, or -\$1. When flipping a fair coin, the probability of heads is the same as the probability of tails, which is 0.5 Therefore,

E = 0.5 (+\$1) + 0.5 (-\$1) = 0.

The expected value of my return is 0. I should expect to neither win nor lose in the long run. In gambling terms, this is “even odds.” I’ll go through some coin flip game scenarios soon. Before that, I ’ d like to calculate a few more expected values.

Consider a roulette wheel. There are 38 slots. Each slot contains one of the following numbers:

0, 00,1,2,3,... ,36,37,38.

There are many different bets that can be placed, but I’ll only consider one simple bet right now. You can bet that the ball will fall into an odd-numbered slot. This would be any of the slots 1, 3, 5, 7, ..., 33, 35, 37. Bet \$1. If the ball falls into any of the odd-numbered slots, you get back \$2, and you’ve won a dollar. If the ball falls into any of the even-numbered slots or the 0 or 00 slots, you’ve lost your dollar.

Since there are 38 slots, 18 of which are odd-numbered slots, your probability of winning is 18/38 and your probability of losing is 20/38. The expected value of your winnings on a \$1 bet is therefore

18 20 2

E = —(+\$1)+—(-\$1) =----------- (+\$1) = -\$0.053.

38 38 38

The expected return is negative. In gambling terms, the odds are against you.

Before examining just what this means, I’d like to present one more expected value calculation. Suppose that a local charity group has received a donation of a fancy TV that’s worth \$2,100. It plans to raffle this TV off. It will sell only 2,000 tickets at \$1 per ticket. What’s the expected value of a ticket purchase?

I f 2,000 tickets are being sold and you buy one ticket, your probability of winning is 1/2,000 = 0.0005. Your probability of losing is 1 - 0.0005 = 0.9995. The expected value of your return on this gamble is

E = 0.0005 (+\$2,100) + 0.9995 (-\$1) = \$1.05 - \$0.9995 = +\$0.0505.

This expected value is positive. The odds are with you on this one.

Summarizing the results of the above three examples, I have a \$1 bet on:

1. a coin flip game with an expected return of 0,

2. a bet on getting an odd number in a spin of a roulette wheel with an expected return of -\$0.053, and

3. a raffle ticket for a \$2,100 TV set with an expected return of +\$0.0505.

These are three gambling opportunities with even odds, odds against me, and odds with me, respectively.

In order to study these three gambling games, I need to look not only at the probability of winning and the expected value of winning, but also at a very impor­tant third factor—how much money I have in my pocket when I start placing my bet(s).

First, let’s look at the coin flip game. The expectation value is 0. You’re as likely to win as you are to lose—or maybe not. The probability of getting heads (or tails) on a given coin flip is 1/2. Winning money with this game is a different story. If the game is to place a bet, flip a coin some number of times, and if you get more heads than tails then you win the bet, otherwise you lose the bet, then, you’re as likely to win as you are to lose. If the game is as described above, you win \$1 for heads, you lose \$1 for tails; we have to look into some details.

Figure 14.1 is a “tree” of possible scenarios when you start out with \$1 in your pocket. Each circle represents a coin flip. The number inside the circle is the amount in your pocket when you bet \$1 and flip the coin. You have a 50% probability (even odds) of winning (W) or losing (L). If you lose (the arrow to the left), you have no money left; you can’t bet again and you must quit. If you win flip #1, you have \$2 and you can continue to bet.

The figure shows that the number of possible scenarios increases with each level of flip. At the first flip, there’s a 50% chance of being wiped out and a 50% chance of increasing your holding to \$2. At the second flip, you can’t get wiped out; your

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bankroll goes either down to \$1 or up to \$3. At the third flip, things start to get busy. Notice that at every flip level, there is a path to getting wiped out.

If you were to start with \$2 in your pocket, the same chart works—j ust put yourself immediately at the flip #2 position and continue onward.

Figure 14.2 shows the total probability of getting wiped out versus the number of times you flip the coin (and bet \$1). The more money you start out with, the lower your probability of getting wiped out. All of the curves shown, however, are increas­ing with the number of coin flips.

Figure 14.3 is the same as Figure 14.2 except that the probability of getting wiped out for up to 2,500 coin flips is shown. Starting out with more money helps, but no matter how much you start out with, sooner or later you will get wiped out. These curves assume that every time you win a dollar, you put it in your wallet and use it to continue betting. If you have to pull some money out for luxuries (dinner), then the probability of your getting wiped out sooner increases.

Another way of looking at this is to calculate the probability of your doubling your money before you get wiped out. Starting with \$1, this is easy—you flip the coin and you’ve either doubled your money or you’re wiped out. Interestingly, this is always the case. It doesn’t matter how much money you start with. You place your dollar bet, flip the coin, and continue. The probability of your getting wiped out before you double your money stays at 50%. In other words, unless you like dragging things out, you might as well bet all of your money on one flip of the coin.

The probability of tripling your money before getting wiped out is 33.3% (1/3), the probability of quadrupling it is 25%, and so on.

Now let’s go back to the roulette wheel. This time it’s easy to reach a conclu­sion. The probability of winning is less than 50%. The expected value of return on a bet is negative. This can’t possibly be a way to earn a living.

 Figure 14.3 Probability of getting wiped out versus the number of coin flips—zoom out.

If there were a roulette wheel with 2,000 slots and you bet on any one of the slots, eventually you’d win. If you do this 1,000 times, then the probability of your winning at least once is about 39%. If you do it 2,000 times, then your probability jumps to 63%; for 5,000 times, it jumps to about 92%. The expected values here are a bit messy to calculate because if you bet 2,000 times, there’s a probability of your winning once, a probability of your winning twice, and so on. Remember, however, that the expected value is always negative. No matter how much you start out with, you should expect to eventually be wiped out.

Finally, the TV raffle. Here, the probability of winning is very small (1/2,000), but the expected value of return is positive. If the expected value of return is posi­tive, shouldn’t you be able to use this as an income stream in the long run?

In the raffle example, the amount of money you start with in your pocket doesn’t just warp the shape of a curve while still leading to the same conclusion; it can actu­ally force different conclusions. First, suppose you have only \$1. You buy a raffle ticket. Since the probability of winning is only 1/2,000, you almost invariably lose.

The raffle is different from the imaginary roulette wheel. Each ticket has a 0.005 probability of winning, and each ticket will pay \$2,100 for a \$1 bet. The roulette wheel never has to pay off. The raffle always has to pay off. With the raffle, if you could rush in early and buy all 2,000 tickets, then you walk away with the prize— there’s no probability to calculate.

The bottom line here seems to be that in a roulette wheel gambling world where there is no required payoff, not only would you need a positive expected value of return, but you would also need enough starting money in your pocket to ride out the variations in order to stick around for a while.

In the real raffle world, there are times when the expected value is positive. The example of a donated raffle prize that’s worth more than the sum of the cost of all the raffle tickets is such an example. If you buy a small fraction of the tickets sold, you’ll probably lose your money. If you can buy all, or almost all, the tickets, you’ll probably get a prize worth more than you paid.

Unfortunately, in most raffles and lotteries, the expected value of return is nega­tive. It’s not impossible to win (someone always wins), but it’s simply a very poor way to spend your money for anything other than charity donations and possibly a little entertainment.

In real roulette wheel and slot machine gambling games, the expected value of return is negative. Occasionally, someone wins. Gambling casinos with a few hundred busy slot machines in one room often have each machine ring some loud bells when a player wins. The resulting cacophony makes it sound like winning is to be expected. Think about it. If the casino didn’t see a positive expected value of return—equivalent to a negative expected value of return for the players—would the casino still be there week after week?

Also, assuming everything is honest, there are no systems. There is no such thing as a slot machine that’s “due” to pay off because slot machines, like flipped coins, have no memory. There are no good lottery numbers. There are no bad lottery numbers. There are no lucky numbers. For a five-digit number, 11111 is as good as 12345 is as good as 42296 is as good as the number that won last week. People used to dealing in probabilities and random events often refer to lotteries as a “tax on stupidity.”

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