Take two problems:

Andrew is 35, and the probability he will be alive in 10 years is .72. Ellen is 35, for her, .92. Assuming these are independent, what is the probability they both will be alive in 10 years?

Answer: .66, Method: Use the multiplication rule .72*.92 = .66

Suppose your street has two traffic lights. The chance that the first light is red is .40, and the second light .30. The chance of them being red at the same time is .10. What is the the probability that neither light is red?

Answer: .40, Method: Subtract .1 from .4 and .3, and then subtract all 3 from 1.0 = .4

**Why doesn't the multiplication rule work for #2?** In other words, why can't I multiply .40 and .30 to get .12? And then furthermore, if .12 is the chance they will both be red at the same time shouldn't (1-.12)=.88 be the chance that neither is on?

#### Best Answer

Let's draw pictures in which *regions* depict *events* (such as "the first light is red") and their *areas* are proportional to the probabilities of those events. Taking care to show areas accurately extends the Venn diagram metaphor in a useful quantitative way.

For the traffic light problem, I will divide a unit square (representing the total probability) into four parts. The left-right division will reflect the possibilities for the first light (set to red at the left, non-red at the right) and the top-bottom division will reflect the possibilities for the second light (red at the bottom, non-red at the top).

**In the left figure**, the divisions have been made in a 40-60 ratio and a 30-70 ratio, respectively. Where the red rectangle (of width 40%) and blue rectangle (of height 30%) intersect they form a purple rectangle of area 30% * 40% = 0.3 * 0.4 = 12% of the total area. *Independent events can always be drawn in this way as separate overlapping rectangles.* (When you think about what this means–overlapping rectangles are a geometric way to multiply quantities–it becomes clear that this is the very definition of independence.)

**The right figure** shows the actual information in the problem, which tells us the purple rectangle has an area of only 10% and asks us to find the area *not* covered by *either* rectangle: the white portion to the upper right, depicting the event "Light 1 is not red and light 2 is not red." This indicates *lack* of independence: now it takes more than just two rectangles to carve up the square correctly. (There's more than one way to do this. For instance, I could have left the blue rectangle alone and adjusted the two halves of the red rectangle, making the bottom skinnier and–to keep its total area at 40%–the top fatter. Either way works.)

### Solution

Starting with the 10% purple rectangle, notice that the rest of the blue rectangle (at the right) has to include the remaining 20% = 30% – 10% of the time the second light is red. Similarly, the rest of the vertical red rectangle has to include the remaining 30% = 40% – 10% of the time the first light is red. This gives three rectangles of known area: 10%, 20%, and 30%. They sum to 60%. Consequently, because the sum of all areas must be 100%, **the white area is 100% – 60% = 40%**. This represents the probability to be found.

### Comments

The white region in the first (left) figure is that of a rectangle with base 60% = 100% – 40% and height 70% = 100% – 30%, whence its area is 0.6 * 0.7 = 42% (and not 40%).

This area = probability method extends to more than two criteria: at What is the probability that this person is female? it is used to analyze a problem with three separate criteria.

Any two-by-two contingency table can (and usually should) be visualized this way, after translating its counts into frequencies relative to the total.

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