1.7 Independence and Compound Probability
When the outcome of the previous event does not affect the outcome of the subsequent event, they are called Independent events. Say, the outcome of rolling a die. No matter, how many times you roll the die, the outcome of a roll will always be independent of any previous or subsequent rolls. Similarly, flipping a coin is another example of independent events since the outcome of trial one will not impact the outcome of trial two. Independent events can lead to Compound Probability, which can be defined as the probability of two or more Independent events occurring. It can be calculated by multiplying the probability of the first independent event by the probability of the second independent event.
The compound probability of independent events and occurring can be calculated as:
Example 1.29. Consider an experiment where we want to find the probability of rolling a 4 on a black die and rolling a 5 on a red die.
We can infer the problem in terms of events:
- () = Rolling a 4 on a black die
- () = Rolling a 5 on a red die
- () = Compound probability that we need to find
Let's start by finding the probabilities of the two events.
There is just one 4 (number of outcomes we are looking for) in the die, while six numbers (number of total possible outcomes) in total in the die. Thus, the probability of rolling a 4 on a Black die is:
Similarly, the probability of rolling a 5 on a Red die is:
Now, to get the probability of events A and B, we simply multiply the outcomes of the two probabilities (as this is a case of Intersection). In our case, there is just one way in which we can get a 4 or 5 in a die, while there can be 36 possible outcomes in total for two dice ({1,1}, {1,2}, {1,3},.....{6,6}). Thus, we get the probability of as:
So far we have talked about independent events. However, what if the outcome of one event influences the outcome of another event? What terminology is used to describe such events, and how can we determine the probability associated with them? The next section explores these questions in-depth and offers a thorough understanding of the topic.