Probability
The probability of something is a number between 0 and 1 saying how likely it is. Nothing is less likely than impossible or more likely than certain, and everything else falls in between as a fraction: how many outcomes you want, over how many outcomes there are. Written that way, most probability questions turn into fraction questions.
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Writing a chance as a fraction
A bag holds 12 counters and 8 of them are red. What is the chance of drawing a red one?
- Count the outcomes you want: 8 of the counters are red.
- Count all the outcomes there are: 12 counters in the bag.
- Write the chance as 8 out of 12, which is 8/12.
- 8 and 12 both divide by 4, so the chance is 2/3.
Worth knowing. The answer is given in lowest terms, the same as any other fraction. 8/12 and 2/3 are the same chance, but only one of them is finished.
The two ends of the scale
A bag holds 9 counters and none of them are red. What is the chance of drawing a red one?
- None of the 9 counters are red, so the chance is 0 out of 9.
- 0 divided by anything is 0.
- The chance is 0, which is what impossible looks like as a number.
- Had all 9 been red the chance would be 9 out of 9, which is 1 — certain.
Worth knowing. 0 and 1 are the ends of the scale and nothing lies outside them. A chance of 1/2 sits exactly in the middle: as likely to happen as not.
Where it usually goes wrong
- Writing the fraction the other way up, so 8 red out of 12 becomes 12/8 — a probability can never be more than 1.
- Comparing the counters you want against the ones you do not, giving 8/4 instead of 8 out of 12.
- Leaving the fraction unsimplified, which is not wrong arithmetic but is an unfinished answer.
The Professor counts the whole bag first and writes that number underneath. The counters he wants go on top, and then he simplifies — a chance is a fraction like any other.
Practise it
All level pathsQuestions teachers and parents ask
Can a probability be bigger than 1?
No. The most that can happen is that every outcome is the one you want, which is the whole over the whole — 1, or certainty. An answer above 1 nearly always means the fraction was written upside down.
Should a chance be written as a fraction, a decimal or a percentage?
All three say the same thing: 1/2, 0.5 and 50% are one answer in three costumes. The questions here ask for the fraction, because a great many chances — a third, a sixth, a seventh — have no tidy decimal to write.
What is the difference between theoretical and experimental probability?
Theoretical probability is what the counting says should happen; experimental probability is what actually happened when someone tried it. They rarely match exactly, and the more trials there are the closer they usually get. Every question here is theoretical.
When is probability taught?
Probability as a number between 0 and 1 is grade 7 (7.SP.C.5), and using a probability model to find the chance of an event is 7.SP.C.7. Younger children meet the same idea as likelihood language — certain, likely, unlikely, impossible — before any fractions are involved.
Where this shows up
Standards
- 7.SP · Statistics and probability
- CCSS.MATH.CONTENT.7.SP.C.5 — Understand that the probability of a chance event is a number between 0 and 1 expressing how likely the event is to happen.
- CCSS.MATH.CONTENT.7.SP.C.7 — Develop a probability model and use it to find the probabilities of events, comparing the model against what is actually observed.