Relationship between two events: Independence vs. dependence

- Flipping a coin and the weather. They're independent, right? Meaning, that if I flip heads on the coin that doesn't impact the probability that it will rain tomorrow. We know that's true, but how can we prove it? Well, if we can say that the probability of flipping heads and getting rain is equal to the product of the probability of heads and the probability of rain, then we can say they are independent. So, for this sample of 100 coin flips and 100 days, given that the coin is flipped heads, what is the probability that it rained? As you can see, this happened 15 of 100 days, 15%, You can also see that the odds of flipping heads is 50%.

The probability of rain was 30%. So, 50% times 30% gives us 15%. Since they are equal, these events are independent, but we sort of knew that going in. So, let's use a more realistic example. For this example we will use this data set. And again, let's consider two events. Event a, a person lives past 85 years of age. Event b, the person exercised three or more days per week.

As you can see, 130 of the 1,000 people had both events a and b, 13%. The probability that a person lived past 85 years was 25%. 250 of 1,000 people lived past 85. The probability that a person in this group exercised three or more days per week was 24%. 240 of 1,000 people exercised three or more days per week.

25% times 24% gives us only 6%. This means that we should expect 6% of those that live past 85 to exercise three or more days per week, but we came up with 13%. From the data it appears that these two events are dependent. In other words, the data from this group tells us that the more you exercise, the longer you are likely to live. On the other hand, in the previous example, it didn't matter if we flipped heads or tails.

The probability that it could rain did not go up or down. Next time someone tries to tell you that two events are related, test the data using the information we just discussed and see if those events are dependent or independent.