Unit E: Statistics and Probability

Chapter 2: Probability

What are Theoretical Probability and Experimental Probability?


There are two types of probability: theoretical probability and experimental probability. Theoretical probability describes the likelihood of an outcome based on reasoning or logic. It predicts the outcome of an experiment without actually doing the experiment.

Experimental probability is based on what actually occurs when an experiment is performed and results are collected.

In an experiment, many trials are performed to collect data. An event is an outcome of the experiment.

The following steps must be followed to perform an experiment:

  1. Identify the trial.
  2. Perform a minimum of 25 trials.
  3. Record the data from the experiment in a table.

The formula to calculate probability is

probability = number   of   favourable   outcomes total   number   of   possible   outcomes  


Matthew decided to conduct an experiment to determine if the experimental probability of heads appearing on a coin when tossed is equal to the theoretical probability. After Matthew completed the experiment, he recorded the results of each toss in the data table below.

Outcome Frequency
heads 21
tails 9
totals 30




  1. Calculate the theoretical probability of a coin landing heads.

  2. Calculate the experimental probability of a coin landing heads.

  3. Matthew continues to toss the coin an additional 20 times and records the totals in the table below.

    Outcome Frequency
    heads 27
    tails 23
    totals 50




    Calculate the new experimental probability of a coin landing on heads.


  1. The theoretical probability is calculated based on reasoning without using the results of an experiment.

    probability = number   of   favourable   outcomes total   number   of   possible   outcomes = number   of   heads   on   a   coin total   number   sides   on   a   coin = 1 2 = 0 . 5

    The theoretical probability for a coin landing heads is 0.5.

  2. The experimental probability is calculated based on data collected in an experiment.

    probability = number   of   favourable   outcomes total   number   of   possible   outcomes = number   of   heads   tossed total   number   of   tosses = 21 30 = 0 . 7

    The probability based on the experiment is 0.7.

  3. probability = number   of   favourable   outcomes total   number   of   possible   outcomes = number   of   heads   tossed total   number   of   tosses = 27 50 = 0 . 54


    The experimental probability after 50 tosses is 0.54.

The experimental probability can vary greatly from the theoretical probability when a small number of trials are completed. The more trials that are performed, the closer the experimental probability will be to the theoretical probability.

Rahib is conducting an experiment to compare the theoretical probability and experimental probability for tossing numbers on a six-sided die.

Outcome Frequency
1 16
2 20
3 22
4 10
5 18
6 14
total 100




  1. Calculate the theoretical probability of the die landing on a 3.

  2. Calculate the experimental probability of a die landing on a 3.

  3. How does the theoretical probability compare to the experimental probability?

  4. How can the experiment be improved so that the experimental probability is closer to the theoretical probability?

  1. The theoretical probability is calculated based on reasoning without using the results of an experiment.

    probability = number   of   favourable   outcomes total   number   of   possible   outcomes = number   of   sides   that   have   three   dots number   of   sides   on   the   die = 1 6 = 0 . 167
    Recall that three decimals are required when converting to decimal form.

    The theoretical probability for a die landing on a 3 is 0.167.

  2. The experimental probability is calculated based on data collected in an experiment.

    probability = number   of   favourable   outcomes total   number   of   possible   outcomes = number   of   threes total   number   of   rolls = 22 100 = 0 . 22

    The probability based on the experiment is 0.22.

  3. The theoretical probability is lower than the experimental probability.

  4. The experiment can be improved by doing more trials. The more trials that are performed, the closer the experimental probability will be to the theoretical probability.

A spinner is divided into eight sections. John spins the spinner 30 times.


 



  1. Calculate the theoretical probability of the spinner landing on red.

  2. Calculate the theoretical probability of the spinner landing on red or green.

  3. John spun the spinner 30 times and recorded his results.

    Outcome Frequency
    red 9
    green 11
    yellow 4
    blue 5
    total 30




    Calculate the experimental probability of the spinner landing on red.

  4. Calculate the experimental probability of the spinner landing on red or green.


  1. probability = number   of   favourable   outcomes total   number   of   possible   outcomes = number   of   red   sections total   number   of   sections = 3 8 = 0 . 375



    The theoretical probability of the spinner landing on red is 0.375.


  2. probability = number   of   favourable   outcomes total   number   of   possible   outcomes = number   of   red   sections + number   of   green   sections total   number   of   sections = 3 + 2 8 = 5 8 = 0 . 625


    The theoretical probability of the spinner landing on red or green is 0.625.


  3. probability = number   of   favourable   outcomes total   number   of   possible   outcomes = number   of   times   landing   on   red total   number   of   spins = 9 30 = 3 10 = 0 . 3


    The experimental probability of the spinner landing on red is 0.3.


  4. probability = number   of   favourable   outcomes total   number   of   possible   outcomes = number   of   times   landing   on   red + number   of   times   landing   on   green total   number   of   spins = 9 + 11 30 = 20 30 = 2 3 = 0 . 667


    The experimental probability of the spinner landing on red or green is 0.667.