An event and its complement (the event not happening) always add up to 1. If P(rain) = 3/10, then P(no rain) = 1 − 3/10 = 7/10. For two separate events, such as tossing a coin and rolling a dice, draw a table: 2 × 6 = 12 equally likely outcomes. Count the cells where both happen. 'Not both' is every other cell, so take the 'both' cells away from all the cells.
Explore this idea with maths tools
Maths you can move
Build 8 + 5
Tap two gold counters to fill the ten-frame.
8 in the frame + 5 waiting. The total stays the same.
Words to know
- complement
- the event not happening
- independent
- one result does not change the chance of the other
- outcome table
- a grid pairing every result of one event with every result of the other
Worked examples
The probability of rain is 3/10. What is the probability of no rain?
- 1.The two must add to 1 = 10/10.
- 2.10/10 − 3/10 = 7/10.
= 7/10
Toss a coin and roll a dice. P(heads and a six)?
- 1.2 × 6 = 12 outcomes.
- 2.Only 1 cell is heads and a six: 1/12.
= 1/12
Toss a coin and roll a dice. P(NOT both heads and an even number)?
- 1.12 outcomes; heads and even: 1 × 3 = 3.
- 2.12 − 3 = 9, so 9/12 = 3/4.
= 3/4
For parents
Method: Complements add to 1; multiply for both
What to say: What is the opposite of this event? How many cells are in the table, and how many fit?
Common mistakes:
- Gives the probability of the event instead of its complement.
- Writes the complement over the event instead of over all outcomes.
- Gives one half because it either happens or it doesn't.
- Adds the two probabilities when both events must happen.
- Adds the numbers of outcomes to get the sample space.
- Gives the probability of just one of the two events.
- Gives P(both) when the question asks for P(not both).
- Treats 'not both' as 'neither'.
- Takes the complement of one event and ignores the other.