Events that are not independent

Two events are independent if the probability of the first event happening has no impact on the probability of the second event happening. If the probability of one event happening affects the probability of other events happening, then the two events are not independent.

Example

A sock drawer contains 5 white socks and 4 black socks. A sock is taken at random and put on. Another sock is taken and put on. What are the probabilities of the different possible colour combinations?

In this question, a sock is taken and not replaced in the drawer. This means that the next time a sock is picked, one of the socks will be missing. This will affect the remaining probabilities.

For example, if the first sock taken is white, there will only be 4 white socks left out of the remaining 8 socks. So the probability that the second sock is white will be \frac{4}{8}. If instead the first sock taken is black, then there will still be 5 white socks left out of the remaining 8. So the probability that the second white sock is white will now be \frac{5}{8}. See where \frac{4}{8} and \frac{5}{8} are shown on the tree diagram.

Tree diagram of getting white and black socks from 2 picks out of a sock drawer

So the probabilities of the different colour combinations are:

Two white socks \frac{5}{9} \times \frac{4}{8} = \frac{20}{72}

One white sock and one black sock \frac{5}{9} \times \frac{4}{8} + \frac{4}{9} \times \frac{5}{8} = \frac{40}{72}

Two black socks \frac {4}{9} \times \frac {3}{8} = \frac{12}{72}