Probability with a Spinner – Step-by-Step Guidance

Q16. The spinner at the right is spun 1 time. Find each of the following probabilities:

• a) $P(2 \text{ or } 3)$

• b) $P(\text{even or a 3})$

• c) $P(\text{a prime and an even number})$

Background

Topic: Basic Probability (Discrete Sample Space)

This question tests your understanding of probability concepts using a spinner with equally likely outcomes. You are asked to find probabilities for different events based on the numbers shown on the spinner.

Key Terms and Formulas

• Probability ($P$): The likelihood of an event occurring, calculated as $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

• Even numbers: Numbers divisible by 2 (on this spinner: 2, 4, 6, 8)

• Prime numbers: Numbers greater than 1 with only two positive divisors: 2, 3, 5, 7

• "Or" Probability: For mutually exclusive events, $P(A \text{ or } B) = P(A) + P(B)$

• "And" Probability: For independent events, $P(A \text{ and } B) = P(A \cap B)$

Step-by-Step Guidance

1. Identify the total number of possible outcomes. The spinner has 8 sections, so there are 8 possible outcomes.

   $\text{Total outcomes} = 8$

2. For part (a), list the numbers that satisfy "2 or 3". These are two separate outcomes: 2 and 3.

   $\text{Favorable outcomes for (a)} = \{2, 3\}$

3. For part (b), list the numbers that are even or a 3. Even numbers are 2, 4, 6, 8. Add 3 to this list (since it's not even).

   $\text{Favorable outcomes for (b)} = \{2, 4, 6, 8, 3\}$

4. For part (c), determine which numbers are both prime and even. On this spinner, the only even prime is 2.

   $\text{Favorable outcomes for (c)} = \{2\}$

5. Set up the probability formula for each part, but do not calculate the final value yet:

   • (a) $P(2 \text{ or } 3) = \frac{\text{Number of favorable outcomes}}{8}$

   • (b) $P(\text{even or a 3}) = \frac{\text{Number of favorable outcomes}}{8}$

   • (c) $P(\text{prime and even}) = \frac{\text{Number of favorable outcomes}}{8}$

Try solving on your own before revealing the answer!

Final Answers:

• (a) $P(2 \text{ or } 3) = \frac{2}{8} = 0.25$

• (b) $P(\text{even or a 3}) = \frac{5}{8} = 0.625$

• (c) $P(\text{prime and even}) = \frac{1}{8} = 0.125$

Each probability is calculated by counting the favorable outcomes and dividing by the total number of spinner sections.