EXAMPLE 4.40

Probabilities as areas. Generate a random number X between 0 and 1. What is the probability that the first digit after the decimal point will be a 3, a 6, or a 9? The random number X is a continuous random variable whose density curve has constant height 1 between 0 and 1 and is 0 elsewhere. The event that the first digit of X is odd is the union of five disjoint events. These events are

0.30 ≤ X < 0.40

0.60 ≤ X < 0.70

0.90 ≤ X < 1.00

Figure 4.16 illustrates the probabilities of these events as areas under the density curve. Each area is 0.1. Therefore, the union of the three has probability equal to the sum, or 0.3.