Probability and equally likely outcomes
Build a sample space before calculating a chance.
Before you begin
- Fractions
- Counting simple outcomes
Choose an appropriate probability model
Probability lies between zero and one. For a finite set of equally likely outcomes, divide the number of favourable outcomes by the total. The equal-likelihood assumption matters: listing red, blue and green does not make each colour equally likely if a bag contains different numbers of each. Count individual equally likely objects instead.
Use complements and interpret experiments
An event and its complement include all possible outcomes without overlap, so their probabilities add to one. Experimental relative frequency is the number of times an event occurs divided by the number of trials. It can differ from a theoretical probability in a short experiment. A prediction describes a long-run tendency, not a guarantee for the next trial.
Worked example
A fair six-sided die is rolled. Find the probability of a number greater than 4.
Show the worked solution
- The six equally likely outcomes are 1, 2, 3, 4, 5 and 6.
- The favourable outcomes are 5 and 6, so there are two.
- Divide 2 by 6 and simplify.
1/3.
Try it yourself
A bag has 3 red and 7 blue identical counters. What is P(red)?
Common mistakes
- Do not assume outcomes are equally likely without evidence.
What can you explain now?
A fair die is rolled. Find P(not a multiple of 3).
Compare with the explanation
2/3.
Multiples of 3 are 3 and 6, with probability 2/6. The complement is 1 − 2/6 = 4/6 = 2/3.
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Common Core mathematics standards ↗