5 H) Multiple Events - Independent & Dependent
Independent events are where the outcome of one event does not affect the outcome of another event. An example of this is the flipping of a coin. If we flip a coin and it lands on a head, it does not mean that there is a greater or lower chance of the coin landing on a head or a tail on the next flip. Another example would be the rolling of a dice. The first roll of the dice does not affect the probabilities for the second roll of the dice. For example, the probability of rolling a 3 on a fair dice is 1/6 and this probability is the same for every roll of the dice – the probability of rolling a 3 does not change; it is always 1/6.
Dependent events are where the probability of an event occurring is dependent on another event. The best example of this is picking a chocolate out of a box. Let’s suppose that a box contains 4 milk chocolates and 3 white chocolates. We are going to pick two chocolates out of the box and eat them (the eating of them means that the chocolate is not going to be replaced back into the box).
There are two options when we pick the first chocolate out of the bag; we either pick a milk or a white chocolate. Whatever chocolate we pick decreases the number of that chocolate in the bag by 1. It also decreases the total number of chocolates in the bag by 1 as well. We initially had 7 chocolates (4 milk and 3 white). So, after picking one chocolate, the total number of chocolates decreases by 1 and this means that there are 6 chocolates left in the bag. Let’s have a look at the two different scenarios in more detail:
Pick a Milk Chocolate
If we were to pick a milk chocolate, the number of milk chocolates is going to decrease by 1. There were initially 4 milk chocolates, which means that after picking a milk chocolate there is going to be 3 milk chocolates left in the box. This means that for the second pick, the probability of picking a milk chocolate is 3/6, which simplifies down to ½.
No white chocolates were picked out of the box, which means that the number of white chocolates is going to remain the same; it will remain as 3. Therefore, the probability of picking a white chocolate out of the bag for the second pick is 3/6, which simplifies down to ½.
Pick a White Chocolate
If we were to pick a white chocolate, the number of white chocolates left in the box is going to decrease by 1. This means that there will be 2 white chocolates available to be picked for the second pick out of the box. Therefore, the probability of picking a white chocolate for the second pick is going to be 2/6, which simplifies down to 1/3.
No milk chocolates were picked, which means that the number of milk chocolates stays at 4. Therefore, the probability of picking a milk chocolate for the second pick is 4/6, which simplifies to 2/3.
Final Note
So, from this example you can see that the probability of picking a milk chocolate for the second pick is dependent on what was picked in the first pick. When the first pick was a milk chocolate, the probability of picking a milk chocolate on the second pick was ½. But, when the first pick was a white chocolate, the probability of picking a milk chocolate on the second pick was 2/3, which is different, thus meaning that the first pick and the second pick are dependent events.
Dependent and independent events are important especially when we are working with probability trees, which we will see towards the end of this whole probability section.