8.04 Expected outcomes
Predictions based on theoretical probability
Remember that we can calculate the theoretical probability of an event by constructing a fraction like this: \text{Theoretical Probability} = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}
An expected outcome, is the value we expect when we apply the theoretical probability to the number of trials:\text{Expected outcome} = \text{Probability}\times \text{number of trials}
Let's look at an example to help us understand.
Examples
Example 1
260 standard six-sided dice are rolled.
What is the probability of getting an even number on a single roll of a die?
How many times would you expect an even number to come up on the 260 dice?
Example 2
The experimental probability that a commuter uses public transport is 50\%. Out of 500 commuters, how many would you expect to use public transport?
Example 3
A car manufacturer found that 1 in every 4 cars they were producing had faulty brake systems. If they had already sold 5060 cars, how many of those already sold would need to be recalled and repaired?
We can calculate the expected outcome by applying the theoretical probability to the number of trials.\text{Expected outcome} = \text{Probability}\times \text{number of trials}
Experimental and expected frequencies
If you have ever done a probability activity in class, you may have noticed that what you expected to happen was different to what actually happened. When this happens we are comparing the expected outcome or the expected frequency and the experimental (or observed) frequency. Remember frequency means how often an event occurs.
We learned in the first section that the expected frequency is how often we think an event will happen. For example, when we flip a coin, we would expect it to land on heads half the time. So if we flipped a coin 100 times, we would expect it to land on heads 50 times because \dfrac{50}{100} = \dfrac{1}{2}.
The experimental or observed frequency is the number of times an event occurs when we do the experiment. For example, let's say we actually decided to flip a coin 100 times. We expected tails to come up 50 times but it only happened 47 times. We would say that the experimental frequency for getting a tail is \dfrac{47}{100}. Notice that the experimental frequency is different to the expected frequency.
Exploration
Now it's your turn to compare expected and experimental frequencies.
- Before you start playing with the applet, write down how often you would expect the spinner to land on each color (i.e. the expected frequency for each color).
- Click the Spin button. The table will record the experimental frequencies. Why do you think the probabilities change after each spin?
- After you have finished the experiment, compare the expected frequencies to the experimental frequencies. What did you notice?
Experimental frequency does not always equal what you expect it to based on the theoretical probability.
Examples
Example 4
A trial is to be conducted by flipping a coin.
What is the theoretical probability of flipping tails on a coin?
A coin was flipped 184 times with 93 tails recorded.
What is the exact experimental probability of flipping tails with this coin?
The expected frequency is how often we think an event will happen.
The experimental or observed frequency is the number of times an event occurs when we run an experiment.