HSPT: Probability: when the event does not happen
Learn to find the probability that an event does not happen, by counting directly or using the complement.
How do you find the probability that something does not happen?
Short answer
Count the outcomes that are not the event and divide by the total. Or find the probability of the event and subtract it from 1.
Know first: Fractions and simplifying, Counting outcomes
Step 1 of 5
Probability is favorable over total
When every outcome is equally likely, the probability of an event is the number of favorable outcomes divided by the total number of outcomes.
A question that asks for the probability of something not happening has two routes. Count the outcomes that are not the event directly, or use the complement: P(not A) = 1 − P(A). Either way, the total stays on the bottom.
| Event | Favorable outcomes | Probability |
|---|---|---|
| Yellow | 4 | 4/15 |
| White | 6 | 6/15 = 2/5 |
| Black | 5 | 5/15 = 1/3 |
| Not white | 4 + 5 = 9 | 9/15 = 3/5 |
P(white) + P(not white) = 2/5 + 3/5 = 1. An event and its complement always add to 1.
Step 2 of 5
Count what is not the event
Worked example: Not white
A bag has 4 yellow, 6 white and 5 black marbles of the same size. One is picked at random. What is the probability it is not white?
- A3/5
- B2/5
- C3/2
- D1/3
- 1
Count the total
4 + 6 + 5 = 15 marbles.
- 2
Count the favorable outcomes
Not white means yellow or black: 4 + 5 = 9.
- 3
Divide and simplify
9/15 = 3/5.
- 4
Check with the complement
P(white) = 6/15 = 2/5, and 1 − 2/5 = 3/5.
- 5
See why the other choices are there
2/5 is the probability that it is white. 3/2 divides not-white by white, which leaves the total out, and no probability is more than 1. 1/3 is the probability of black only.
Answer
3/5
Wrong: P(not white) = 9/6
Right: P(not white) = 9/15
The bottom of a probability is always the total number of outcomes. Dividing by the other group gives a comparison, and it can even be more than 1.
Step 3 of 5
Keep the total on the bottom
Check yourself · Question 1
A spinner has 8 equal sections numbered 1 to 8. What is the probability of not landing on a multiple of 3?
Answer: B
The multiples of 3 from 1 to 8 are 3 and 6, so 2 sections. That leaves 8 − 2 = 6 sections, and 6/8 = 3/4.
Trap answer: 1/4
Answering the opposite event
- Why it looks right
- Counting the multiples of 3 is the main work, so their probability feels like the answer.
- Why it's wrong
- The question asks for not landing on a multiple of 3. That is 1 − 1/4.
- The right answer
- 3/4
Check yourself · Question 2
A jar has 12 marbles. The probability of picking a blue one is 1/4. How many marbles in the jar are not blue?
Answer: C
Blue marbles: 1/4 of 12 = 3. Not blue: 12 − 3 = 9. Check: 9/12 = 3/4, which is 1 − 1/4.
Common mistake
"Not white means I divide the other marbles by the white ones."
- Why it's tempting
- Not white compares two groups, so a group-to-group fraction seems natural.
- Do this instead
- A probability always puts the total on the bottom. Count the favorable outcomes and divide by every possible outcome.
Step 4 of 5
What to remember
Remember
- Probability = favorable outcomes ÷ total outcomes.
- P(not A) = 1 − P(A), or count the outcomes that are not A directly.
- A probability is never more than 1; an answer above 1 has the wrong bottom number.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
How many outcomes were favorable in the practice question, and what was the total you divided by?