Calculating theoretical probabilities
In a nutshell
In order to calculate theoretical probabilities for events from experiments you need to know all the possible outcomes. An outcome of an experiment is itself an event, but it often is also one of many outcomes that can lead to a certain event occurring.
Theoretical probability
Terms
Trial | An activity or experiment. |
Outcome | A result of a trial. |
Event | A subset of all of the possible outcomes of a trial. |
The theoretical probability of an event A is
P(A)=Total number of trialsNumber of times event occurs
Example 1
Alan rolls a fair die. What is the probability that the die lands on a prime number?
As the outcomes of a fair die are all equal, the probability of landing on any number is 61. The outcomes and their resepective probabilities can be show in a table:
XP(X=x)161261361461561661
The outcomes associated with the event of landing on a prime number is if the die lands on 2,3 or 5. Hence,
P(prime)=P(2,3,5)=P(2)+P(3)+P(5)=61+61+61=21
Counting outcomes
To find the probability of an event, we need to count the number of outcomes that lead to the event occurring. To find the total number of outcomes that lead to multiple mutually exclusive events, you can multiply the number of outcomes together.
Note: Mutually exclusive events are events that cannot occur at the same time. If two events A and B are mutually exclusive, then P(A∩B)=0.
Example 2
A restaurant offers 3 starters, 10 main courses and 5 desserts. How many different outcomes are there for an entire sitting?
Obtaining a starter is an event, as are obtaining main courses and desserts. Hence there are 3×10×5=150 different possible outcomes.