# Calculating Probability & Relative Frequencies

Instructor: Linda Richard

Linda is a National Board Certified math teacher with master's degrees in teaching and applied mathematics.

Learn to model simple situations by calculating probabilities for events with equally likely outcomes. Use these probabilities to make predictions, and evaluate your model by comparing real-life observations to the predictions.

Probability is the likelihood of things happening. Will it rain this weekend? Will the home team win the big game? Do the socks we randomly grabbed this morning match? Math can help model these scenarios and more.

## Number of Trials and the Probability of Outcomes

Say you roll your lucky six-sided die every day at the gym to decide how many pull-ups to do.

experimenttrialoutcomesample spaceevents

If the die is fair, then each number has the same chance of being rolled. So, we say that this experiment has equally likely outcomes. An event's probability in an experiment with equally likely outcomes is defined by this formula, where P(event) means the probability of the event occurring:

Consider the event 'rolling a multiple of 3.' There are 2 outcomes in this event: the numbers 3 and 6. There are 6 total outcomes in the sample space. Plug those numbers into the formula to find the probability of the event occurring.

This fraction simplifies to one-third or about 33.3%, so we can say that there's a 33.3% probability of getting a multiple of 3 on any one roll.

## Relative Frequencies and the Fairness of Outcomes

Now, after a few weeks of gym-going, you have a feeling that you've been getting more 4s, 5s, and 6s than you should. If the die is fair, the probability of rolling, say, a 5 on any given roll is 1 divided by 6, or 16.7%. Another way to think about this percentage is that if we tally the outcomes over many rolls, we expect to see a 5 rolled about 16.7% of the time. When thought of in this way, not as a probability but as a proportion of actual results, the 16.7% is called a relative frequency.

To test whether the die is fair (and that your mathematical model of equally likely outcomes is correct), you can roll the die several times and compare the 16.7% predicted relative frequency to the relative frequency you actually observe. Say you roll the die 30 times and count 7 5s. To calculate the relative frequency of this outcome, divide the number of times it occurred by the total number of trials. That gives 7 divided by 30 or 23.3%.

### Observed Relative Frequencies

This observed relative frequency of 23.3% doesn't exactly match the predicted relative frequency of 16.7%. However, we can't yet conclude that the die isn't fair. An element of chance is at play in all situations involving probability, and real-world results won't usually exactly match the mathematical model. The result of 23.3% could easily have happened just by chance.

To further analyze the model, let's look at the relative frequencies of the rest of the outcomes.

Outcome 1 2 3 4 5 6
Number of occurrences 1 0 1 11 7 10
Relative frequency 3.3% 0% 3.3% 36.7% 23.3% 33.3%

Yikes! Those relative frequencies are dramatically different from the model's prediction of 16.7% for each number. This doesn't prove with 100% certainty that the die isn't fair; it's technically possible that these results could just be from chance. However, it's strong evidence that these outcomes aren't equally likely. The die is likely weighted to give more 4s, 5s, and 6s, than 1s, 2s, and 3s.

With a sigh, you throw away your lucky die and buy a new one. You tally up your first 30 days' worth of results with the new die.

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