Probability - Compound Events

🔑 Key Concepts

• A compound event is made of two or more simple events.
• Independent events do not affect each other, such as spinning the same spinner more than once.
• To find the probability of independent events happening in a row, multiply the probabilities.
• The complement of an event includes all outcomes that are not part of the event.
• Sample spaces can help organize and count possible outcomes.

✏️ Worked Examples

Spinning an Even Number More Than Once

Number Spinner

The spinner has 6 equal sections numbered 1 through 6.

Even Numbers: 2, 4, 6
Odd Numbers: 1, 3, 5

The favorable outcomes for spinning an even number are 2, 4, and 6. Since 3 of the 6 equal sections are even:

\( P(\text{even}) = \frac{3}{6} = \frac{1}{2} \)

Question 1: What is the probability of spinning an even number two times in a row?

Since the spins are independent, multiply the probability of even on the first spin by the probability of even on the second spin.

\( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \)

Answer: The probability of spinning an even number two times in a row is \( \frac{1}{4} \).

Question 2: What is the probability of spinning an even number three times in a row?

Multiply the probability of spinning even on each spin.

\( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \)

Answer: The probability of spinning an even number three times in a row is \( \frac{1}{8} \).

Complement Connection:

The complement of spinning an even number is spinning an odd number. The odd numbers are 1, 3, and 5.

\( P(\text{odd}) = \frac{3}{6} = \frac{1}{2} \)

Notice that the probability of an event and its complement always add to 1: \( \frac{1}{2} + \frac{1}{2} = 1 \).

🧠 Math Vocabulary

• Probability: The likelihood that an event will happen.
• Theoretical Probability: Probability based on reasoning and equally likely outcomes.
• Experimental Probability: Probability based on actual results from an experiment or simulation.
• Compound Event: An event made up of two or more simple events.
• Independent Events: Events where the outcome of one event does not affect the outcome of another event.
• Favorable Outcome: An outcome that satisfies the event being considered.
• Complement: All outcomes that are not part of the event.
• Sample Space: The set of all possible outcomes.

💡 Main Idea

Compound probability helps us find the chance of more than one event happening. When events are independent, such as spinning the same spinner several times, we multiply the probability of each event.

📚 What You Should Already Know

Students should know how to find simple probability, simplify fractions, identify favorable outcomes, and understand that probability compares favorable outcomes to total possible outcomes.

🚀 What Comes Next

Next, students can use sample spaces, tree diagrams, organized lists, and simulations to find probabilities for more complex compound events.

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