Experimental probability: frequency trees
In a nutshell
Experimental probability, or relative frequency, describes how frequently an event occurred in a completed experiment. You can record and organise information to calculate probabilities by building a frequency tree.
Experimental probability
Relative frequency is used for calculating a probability using the outcome of an experiment. The theoretical probability of getting tails from a fair coin is 21. However, if you flip a coin 100 times, you may not actually get tails 50 times.
Relative frequency is calculated by dividing the number of times an event has occurred, by the total number of trials in a given experiment. The more times an experiment is carried out, the more reliable the relative frequency will be.
Relative frequency =total number of trialsnumber of times an event occurred
Example 1
A coin is tossed 10 times. Of these times, the coin lands on heads 3 times and on tails 7 times.
Therefore the relative frequency for each event is:
Relative frequency of headsRelative frequency of tails=103=107
Example 2
The table shows the number of cookies a bakery sold this week.
Based on this data, what is the reasonable estimate of the probability that the bakery will sell more than two dozen cookies the next day?
Remember that:
Relative frequency =total number of trialsnumber of times an event occurred
The event is the bakery selling more than 24 cookies in a day, P(>24).
From the table, this occurred on Saturday (sold 31 cookies) and Sunday (sold 34 cookies). So the bakery sold more than 24 cookies on 2 days in the last week.
Number of times event occurred =2
The data is a week's worth which is 7 days.
Total number of trials =7
Put it all together for your answer.
Relative frequency =72
Frequency trees
Frequency trees are a way of recording and organising information given as frequencies. They show how things can be broken into different categories. They can then be used to calculate probabilities.
Example 3
There are 83 languages students. 44 are in Year 10 and 39 in Year 11. In Year 10, 26 take Spanish and 18 take French. In Year 11, 19 take Spanish and the rest take French.
The frequency tree for this information will look like this:
If a languages student was picked randomly, what is the probability that they study French?
From the frequency tree, you can see that 18 students from Year 10 study French, and 20 Year 11 students study French.
18+20=38
There is a total of 83 languages students. So the probability that a randomly selected student studies French is:
Relative frequency for a student to study French =8338