Our next examination topic is independence. This is an area which causes some difficulty, and we will start with simple examples and build up to more complex situations over a number of steps.
We will start with a definition: two events are independent if the occurrence or non-occurrence of one has no effect on the probability of the other.
Some examples should make this definition clearer:
Suppose we roll an ordinary dice and flip a coin. If event \(A\) is ‘score 6 on the dice’ and event \(B\) is ‘get Heads on the coin’, then \(A\) and \(B\) are independent - the result of the dice roll has no influence on the coin flip, and vice-versa.
Now suppose we have a set of 10 cards, marked with each of the integers 1 - 10, and choose two of the cards at random, without replacement. Let event \(C\) be ‘the first number chosen is even’, and event \(D\) be ‘the second number is even’. Clearly, \(C\) and \(D\) are not independent events - you should be able to verify that the probability of event \(D\) varies, depending on whether \(C\) occurs or not.
We use the notation \(p(X\)|\(Y)\) to denote the probability that event \(X\) occurs, given that \(Y\) has occurred. If \(X\) and \(Y\) are independent, then we can write \(p(X\)|\(Y)=p(X)\).
As a final piece of theory at this stage, we can introduce the familiar product rule for independent events: if \(X\) and \(Y\) are independent events, we can multiply their individual probabilities to find the probability that both events occur:\[p(X\cap Y)=p(X).p(Y)\]
Now let us move on to some typical examination-style questions for this topic. We will start with the simplest case - two identical independent events. In these examples, the second event that we are considering is essentially a repeat of the first. The following questions should make this clear.
Sample examination-style questions
Faiza throws a fair die and writes down the number that comes up. She throws it again and writes down this second number. She than subtracts the second number from the first number.
Copy and complete the downloadable table for this step showing the possible outcomes.
What is the probability that the result of the subtraction is 0?
What is the probability that the result of the subtraction is greater than 4?
What is the probability that the result of the subtraction is less than -3?
The probability that it rains in the morning is 0.2, and there is the same probability of rain in the afternoon, independent of what happened in the morning.
Draw a (two-stage) probability tree diagram to show the possible outcomes of the day’s weather, and find the probability associated with each outcome.
What is the probability that it rains at some point in the day?
Use the comments to discuss or clarify any of the material introduced here, and to share your answers to the questions. In the next step we will present our solutions, and give you an opportunity to discuss teaching approaches for this topic.