Determine probabilities of complementary events

The complement of an event consists of all outcomes in the sample space for which the event does not occur, so the event and its complement are mutually exclusive and exhaustive. Using $P(A^c)=1-P(A)$ , learners determine probabilities expressed as fractions, decimals, or percentages, including “at least one” situations by considering “none”; this does not require independence or conditional probability.

Detailed Explanation: 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 $40\%$ .

Learn by doing: Determine probabilities of complementary events

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Probability - Coins (2), Not All Specific, To Fraction

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