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For independent events, the probability of both of them happening is the product of the individual probabilities for the events.

      You may find it amazing, but that’s all you really need to know about probability.

      Playing a game

      Now that you’re familiar with coin flipping, I’d like to challenge you to a game. I’ll flip a coin again and again until it turns up heads. If it turns up heads on the first flip, I’ll give you £2. If it turns up heads for the first time on the second flip, I’ll give you £4. If it turns up heads for the first time on the third flip, I’ll give you £2³ = £8. And if it turns up heads on the nth flip I’ll give you £2ⁿ. How much are you prepared to pay me to play this game?

      To work out how much you may win, you need to calculate some probabilities. The probability of the coin turning up heads is always 0.5, and the probability the coin turns up tails is also 0.5. If heads appears first on the nth (say, third) flip, then all previous flips must have been tails. The probability of that is 1/2^(n-1) (so 0.5² if n = 3). You must now multiply again by 0.5 to get the probability of heads on the nth flip preceded by tails on the previous (n–1) flips. This works out as 1/2ⁿ (so 0.5 × 0.5² = 0.5³ if n = 3).

      So, heads turn up for the first time on the nth flip with probability 1/2ⁿ. If heads turns up first on the nth flip, then you win £2ⁿ. The total expected pay-off (the amount, on average, you receive for winning) is then:

      £2/2 + £2²/2² + £2³/2³ + …

      However, this is just a series of 1s going on forever that adds up to infinity. So, then, would you pay me your life savings to play this game in the hope of a staggering return? If heads came up first, you may be disappointed at receiving a mere £2 for your savings; but if you had to wait a long time for heads to turn up but eventually it did and you were due a substantial pay off, I may not be able to pay out your winnings. I don’t think that the Central Bank would print large amounts of money to help me out. This is an extreme example in which an unlikely event plays a significant role. You may notice a spooky similarity to certain recent events in financial markets even though this game was invented several hundred years ago.

      Flipping more coins

      Another fun experiment with a coin is to keep on flipping it again and again to see how many times heads comes up. Sometimes heads follows tails and at other times there can be long series of either heads or tails.

      

During long sequences of heads or tails, you can easily believe that you have a higher than average probability of the other side turning up to even things up a bit. This gambler’s fallacy, however, isn’t valid. The coin has no memory. On each flip, the probability of heads remains 0.5, as does the probability for tails.

      An important idea that comes out of experimenting with gambling games is the Law of Large Numbers. It states that the average result from a large number of trials (such as coin tossing) should be close to the expected value (0.5 for tossing heads) and will become closer as more trials are performed. I’ll show you how this works.

If H_n is the total number of heads (for example, 4) in the first n tosses (for example, 8) then H_n/n should tend towards 0.5 (so, 4/8 = 0.5). Figure 2-1 graphs 1,000 coin tosses.

      © John Wiley & Sons, Ltd.

      FIGURE 2-1: Convergence of the proportion of tossed coins landing heads up.

      The chart fluctuates less and less after more coin flips and the fraction of heads converges (gets closer and closer) towards 0.5. This is an example of the Law of Large Numbers. You’d be surprised though at how many tosses it takes for the chart to settle down to the expected average.

      I examine this further by plotting H_n – n/2 where n/2 is the expected number of heads after n tosses. The line in Figure 2-2 wanders about and shows that convergence isn’t good. It’s disconcerting that although the fraction of heads tossed tends towards 0.5 in relative terms, in absolute terms, the number of heads can wander further and further away from the expected value of n/2. You may have guessed already that this unstable sequence, called a random walk, can be used as a model for how share prices change with time.

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