# 1.18. Theoretical vs. experimental probabilities

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Mathematicians define probability as:

Probability of an event | = | Number of ways in which the event can happen |

Total number of (equally likely) possible outcomes in the situation |

**Experimental probability**

Experimental probability refers to the probability of an event occurring when an experiment was conducted. In such a case, the probability of an event is being determined through an actual experiment. Mathematically,

Experimental probability | = | Number of event occurrences |

Total number of trials |

For example, if a die is rolled 6000 times and the number ‘5’ occurs 990 times, then the experimental probability that ‘5’ shows up on the die is 990/6000 = 0.165.

**Theoretical probability**

On the other hand, theoretical probability is determined by noting all the possible outcomes theoretically, and determining how likely the given outcome is. Mathematically,

Theoretical probability | = | Number of favourable outcomes |

Total number of outcomes |

For example, the theoretical probability that the number ‘5’ shows up on a die when rolled is 1/6 = 0.167. This is because of the 6 possible outcomes (die showing ‘1’, ’2’, ’3’, ’4’, ’5’, ’6’), only 1 outcome (die showing ‘5’) is favourable.

**Note:**

As the number of trials keeps increasing, the experimental probability tends towards the theoretical probability. To see this, the number of trials should be sufficiently large in number.

Experimental probability is frequently used in research and experiments of social sciences, behavioural sciences, economics and medicine.

In cases where the theoretical probability cannot be calculated, we need to rely on experimental probability.

For example, to find out how effective a given cure for a pathogen in mice is, we simply take a number of mice with the pathogen and inject our cure. We then find out how many mice were cured and this would give us the experimental probability that a mouse is cured to be the ratio of number of mice cured to the total number of mice tested. In this case, it is not possible to calculate the theoretical probability. We can then extend this experimental probability to all mice.

It should be noted that in order for experimental probability to be meaningful in research, the sample size must be sufficiently large.

Although experimental probability plays a very important role in our everyday lives, mathematicians prefer to see if it is possible to make predictions based on their knowledge of the situation (theory), rather than experimental evidence.

**Example 1:**

In a spinner, we could divide the circle into 360 equally sized segments. If we spin the spinner, we could imagine that the arrow will always end up in one of the marked segments (one of the 360). We must also assume that the spinner is not biased – the arrow can stop anywhere.

This would mean that it is equally like that the arrow can stop in any one of the 360 positions

From our spinner, we can see that Red covers 180 positions. So we can calculate

Probability of a Red outcome | = | 180 | = | 1 | = | 0.5 | = | 50% |

360 | 2 |

because 180 of the possible outcomes are red (the number of ways in which we can get a red) and there are 360 equally likely possible outcomes.

We can then say that the theoretical probability of a red is one half. This means that we would expect / predict one in two outcomes of every large set of trial to be red, using this spinner.

From our spinner, we can see that Green (and also Blue) covers 90 positions each. So we can calculate

Probability of a Green outcome | = | 90 | = | 1 | = | 0.25 | = | 25% |

360 | 4 |

because 90 of the possible outcomes are green (or blue) (the number of ways in which we can get a green [or a blue]) and there are 360 equally likely possible outcomes.

We can then say that the theoretical probability of a green is one quarter. This means that we would expect / predict one in four outcomes of every large set of trial to be green (or blue), using this spinner.

**Note:**

The theoretical probability does not say that exactly one outcome in exactly 2 trials will always be red. It predicts the ratio of red outcomes to all outcomes as the experiment is repeated many times.

The theoretical probability cannot tell us what the outcome of the next trial will be in a series of experiments. If we get a red on the first trial, we can just as easily get another red on the second trial.

Let’s look at an example:

**Problem: **

A spinner has 4 equal sectors coloured yellow, blue, green and red. What are the chances of landing on blue after spinning the spinner? What are the chances of landing on red?

**Solution: **

The chances of landing on blue are 1 in 4, or one fourth.

The chances of landing on red are 1 in 4, or one fourth.

This problem asked us to find some probabilities involving a spinner. Let’s look at some definitions and examples from the problem above:

Definition | Example |

An experiment is a situation involving chance or probability that leads to results called outcomes. | In the problem above, the experiment is spinning the spinner. |

An outcome is the result of a single trial of an experiment. | The possible outcomes are landing on yellow, blue, green or red. |

An event is one or more outcomes of an experiment. | One event of this experiment is landing on blue. |

Probability is the measure of how likely an event is.P(A) = The Number Of Ways Event A Can Occur The total number Of Possible Outcomes The probability of event A is the number of ways event A can occur divided by the total number of possible outcomes. | The probability of landing on blue is one fourth. |

Let’s take a look at a slight modification of the problem above:

**Experiment 1: **

A spinner has 4 equal sectors coloured yellow, blue, green and red. After spinning the spinner, what is the probability of landing on each colour?

**Outcomes: **

The possible outcomes of this experiment are yellow, blue, green, and red.

**Probabilities: **

P(yellow) | = | # of ways to land on yellow | = | 1 |

total # of colours | 4 | |||

P(blue) | = | # of ways to land on blue | = | 1 |

total # of colours | 4 | |||

P(green) | = | # of ways to land on green | = | 1 |

total # of colours | 4 | |||

P(red) | = | # of ways to land on red | = | 1 |

total # of colours | 4 |

**Experiment 2: **

A single 6-sided die is rolled.

- What is the probability of each outcome?
- What is the probability of rolling an even number?
- Of rolling an odd number?

**Outcomes: **

The possible outcomes of this experiment are 1, 2, 3, 4, 5 and 6.

**Probabilities: **

P(1) = # of ways to roll a 1 = 1 total # of sides 6 P(2) = # of ways to roll a 2 = 1 total # of sides 6 P(3) = # of ways to roll a 3 = 1 total # of sides 6 P(4) = # of ways to roll a 4 = 1 total # of sides 6 P(5) = # of ways to roll a 5 = 1 total # of sides 6 P(6) = # of ways to roll a 6 = 1 total # of sides 6 P(even) = # ways to roll an even number = 3 = 1 total # of sides 6 2 P(odd) = # ways to roll an odd number = 3 = 1 total # of sides 6 2 |

Experiment 2 illustrates the difference between an outcome and an event. A single outcome of this experiment is rolling a 1, or rolling a 2, or rolling a 3, etc. Rolling an even number (2, 4 or 6) is an event, and rolling an odd number (1, 3 or 5) is also an event.

**Note:**

In Experiment 1 the probability of each outcome is always the same. The probability of landing on each colour of the spinner is always one fourth.

In Experiment 2, the probability of rolling each number on the die is always one sixth. In both of these experiments, the outcomes are equally likely to occur.

Let’s look at an experiment in which the outcomes are not equally likely:

**Experiment 3: **

A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles.

If a single marble is chosen at random from the jar, what is the probability of choosing:

- A red marble?
- A green marble?
- A blue marble?
- A yellow marble?

**Outcomes: **

The possible outcomes of this experiment are red, green, blue and yellow.

**Probabilities:**

P(red) | = | # of ways to choose red | = | 6 | = | 3 |

total # of marbles | 22 | 11 | ||||

P(green) | = | # of ways to choose green | = | 5 | ||

total # of marbles | 22 | |||||

P(blue) | = | # of ways to choose blue | = | 8 | = | 4 |

total # of marbles | 22 | 11 | ||||

P(yellow) | = | # of ways to choose yellow | = | 3 | ||

total # of marbles | 22 |

The outcomes in this experiment are not equally likely to occur. You are more likely to choose a blue marble than any other colour. You are least likely to choose a yellow marble.

**Experiment 4:**

Choose a number at random from 1 to 5. What is the probability of each outcome? What is the probability that the number chosen is even? What is the probability that the number chosen is odd?

**Outcomes:**

The possible outcomes of this experiment are 1, 2, 3, 4 and 5.

**Probabilities:**

P(1) | = | # of ways to choose a 1 | = | 1 |

total # of numbers | 5 | |||

P(2) | = | # of ways to choose a 2 | = | 1 |

total # of numbers | 5 | |||

P(3) | = | # of ways to choose a 3 | = | 1 |

total # of numbers | 5 | |||

P(4) | = | # of ways to choose a 4 | = | 1 |

total # of numbers | 5 | |||

P(5) | = | # of ways to choose a 5 | = | 1 |

total # of numbers | 5 | |||

P(even) | = | # of ways to choose an even number | = | 2 |

total # of numbers | 5 | |||

P(odd) | = | # of ways to choose an odd number | = | 3 |

total # of numbers | 5 |

The outcomes 1, 2, 3, 4 and 5 are equally likely to occur as a result of this experiment. However, the events even and odd are not equally likely to occur, since there are 3 odd numbers and only 2 even numbers from 1 to 5.

**Summary: **

The probability of an event is the measure of the chance that the event will occur as a result of an experiment. The probability of an event A is the number of ways event A can occur divided by the total number of possible outcomes. The probability of an event A, symbolised by P(A), is a number between 0 and 1, inclusive, that measures the likelihood of an event in the following way:

- If P(A) > P(B) then event A is more likely to occur than event B.
- If P(A) = P(B) then events A and B are equally likely to occur.