Events that are not independent – Higher
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 socks being different colours?
In this example, 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.
In this tree diagram if a white sock has been selected and put on there are no longer 5 white socks in the drawer. There are only 4. The total number of socks has also gone from 9 to 8. Similarly if a black sock has been selected and put on there are no longer 4 black socks in the drawer. There are only 3.
So the probabilities of the different sock pairings are:
two white socks \(\frac{5}{9} \times \frac{4}{8} = \frac{20}{72} = \frac{5}{18}\)
one white sock and one black sock in either order\(\frac{5}{9} \times \frac{4}{8} + \frac{4}{9} \times \frac{5}{8} = \frac{40}{72} = \frac{5}{9}\)
two black socks \(\frac {4}{9} \times \frac {3}{8} = \frac{12}{72} = \frac{1}{6}\)