"Not" means 1 minus
A path has 10 equal stones. 4 are wet and the rest are dry. You step on one stone without looking. What is the chance it is dry?
Step 1 of 4
- 10 stones. 4 are wet.
- P(wet) = 4/10 = 2/5.
- Every other stone is dry. P(dry) = 1 − 2/5 = 3/5.
- Check: 2/5 + 3/5 = 5/5 = 1.
Dry stones are 6 of 10. The chance is 3/5.
The rule
P(not E) = 1 − P(E)
- Not E — every outcome that is not in E. The book calls it the complement of E.
- E and "not E" share nothing. Together they cover everything. So P(E) + P(not E) = 1.
- Elementary event — an event with just one outcome. Take all elementary events of one pick. Their chances add up to 1.
When it saves time
A spinner has 4 equal parts, numbered 1 to 4. Spin it two times. What is P(at least one 4)? At least one 4 means one 4 or two 4s.
- All results: 4 × 4 = 16 pairs.
- Each spin has 3 results that are not 4. So no 4 at all: 3 × 3 = 9 pairs. P(no 4) = 9/16.
- P(at least one 4) = 1 − 9/16 = 7/16.
Check by listing all 16 pairs. 7 pairs have a 4.
| E | P(E) | P(not E) |
|---|---|---|
| a multiple of 3 (3, 6, 9) | 3/10 | 1 − 3/10 = 7/10 |
| more than 6 (7, 8, 9, 10) | 4/10 = 2/5 | 1 − 2/5 = 3/5 |
| a slip showing 12 | 0 | 1 − 0 = 1 |
Now try it. Slide the number of wet stones. Watch both chances change.
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