Skill: Determine probabilities of complementary events

The complement of an event is the event that it does not happen.

If an event is called $A$, its complement is written $A^c$. The probabilities of an event and its complement always add to $1$:

$$P(A^c)=1-P(A)$$

Example

A player has a probability of $\frac{3}{5}$ of scoring at least one point in a game. What is the probability that the player scores no points?

Step 1: Identify the event and its complement.

• Event $A$: scoring at least one point
• Complement $A^c$: scoring no points

These two possibilities cover every outcome, so they are complements.

Step 2: Use the complement rule.

$$P(A^c)=1-P(A)$$

Substitute $P(A)=\frac{3}{5}$:

$$P(A^c)=1-\frac{3}{5}$$

Step 3: Subtract.

$$P(A^c)=\frac{5}{5}-\frac{3}{5}=\frac{2}{5}$$

So, the probability of scoring no points is

$$\boxed{\frac{2}{5}}$$

This is also $0.4$ or $40\%$.