Before discussing photons, we need to introduce some ideas about probability. Imagine throwing darts at a dart board while blindfolded. FIGURE 39.2 shows how the board might look after your first 100 throws. From this information, can you predict where your 101st throw is going to land? We’ll assume that all darts hit the board.

FIGURE 39.2 One hundred throws at a dart board.

No. The position of any individual dart is unpredictable. No matter how hard you try to reproduce the previous throw, a second dart will not land at the same place. Yet there is clearly an overall pattern to where the darts strike the board. Even blindfolded, you had a general sense of where the center of the board was, so each dart was more likely to land near the center than at the edge.

Although we can’t predict where any individual dart will land, we can use the information in Figure 39.2 to determine the probability that your next throw will land in region A or region B or region C. Because 45 out of 100 throws landed in region A, we could say that the odds of hitting region A are 45 out of 100, or 45%.

Now, 100 throws isn’t all that many. If you throw another 100 darts, perhaps only 43 will land in region A. Then maybe 48 of the next 100 throws. Imagine that the total number of throws $N_{\text{tot}}$ becomes extremely large. Then the probability that any particular throw lands in region A is defined to be

In other words, the probability that the outcome will be A is the fraction of outcomes that are A in an enormously large number of trials. Similarly, $P_{\text{B}}=N_{\text{B}}\text{/}{N}_{\text{tot}}$ and $P_{\text{C}}=N_{\text{C}}\text{/}{N}_{\text{tot}}$ as $N_{\text{tot}}\to \infty$. We can give probabilities as either a decimal fraction or a percentage. In this example, $P_{\text{A}}\approx 45\%\text{,}{P}_{\text{B}}\approx 35\%$, and $P_{\text{C}}\approx 20\%$. We’ve used $\approx$ rather than $=$ because 100 throws isn’t enough to determine the probabilities with great precision.

What is the probability that a dart lands in either region A or region B? The number of darts landing in either A or B is $N_{\text{A or B}}=N_{\text{A}}+N_{\text{B}}$, so we can use the definition of probability to learn that

That is, the probability that the outcome will be A or B is the sum of $P_A$ and $P_B$. This important conclusion is a general property of probabilities.

Each dart lands somewhere on the board. Consequently, the probability that a dart lands in A or B or C must be 100%. And, in fact,

Thus another important property of probabilities is that the sum of the probabilities of all possible outcomes must equal 1.

Suppose exhaustive trials have established that the probability of a dart landing in region A is $P_{\text{A}}$. If you throw $N$ darts, how many do you expect to land in A? This value, called the expected value, is

The expected value is your best prediction of the outcome of an experiment.

If $P_{\text{A}}=0.45$, your best prediction is that 27 of 60 throws (45% of 60) will land in A. Of course, predicting 27 and actually getting 27 aren’t the same thing. You would predict 30 heads in 60 flips of a coin, but you wouldn’t be surprised if the actual number were 28 or 31. Similarly, the number of darts landing in region A might be 24 or 29 instead of 27. In general, the agreement between actual values and expected values improves as you throw more darts.