There is a spinner with 12 equal areas, numbered 1 through 12. If the spinner is spun one time,
what is the probability that the result is a multiple of 2 or a multiple of 3 ?
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There is a spinner with 12 equal areas, numbered 1 through 12. If the spinner is spun one time, what is the probability that the result is a multiple of 2 or a multiple of 3 ? Answer

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1. The spinner has 12 equally likely outcomes, numbered from 1 to 12.

2. We need to find the probability that the outcome is a multiple of 2 or a multiple of 3.

3. First, count the multiples of 2:
   - The multiples of 2 in the range are: $$2, 4, 6, 8, 10, 12$$.
   - There are 6 multiples of 2.

4. Next, count the multiples of 3:
   - The multiples of 3 in the range are: $$3, 6, 9, 12$$.
   - There are 4 multiples of 3.

5. Some numbers (namely $$6$$ and $$12$$) are counted twice because they are multiples of both 2 and 3. Count these intersection numbers:
   - The numbers that are multiples of both 2 and 3 (i.e., multiples of 6) are: $$6, 12$$.
   - There are 2 such numbers.

6. Use the inclusion-exclusion principle:
   - The total favorable outcomes is:
     $$
     \text{Favorable outcomes} = (\text{Multiples of }2) + (\text{Multiples of }3) - (\text{Multiples of both }2 \text{ and }3)
     $$
     $$
     = 6 + 4 - 2 = 8
     $$

7. Finally, divide the number of favorable outcomes by the total number of outcomes:
   $$
   P(\text{multiple of 2 or 3}) = \frac{8}{12} = \frac{2}{3}
   $$

Thus, the probability is $$ \frac{2}{3} $$.
The probability of spinning a multiple of 2 or 3 is $$ \frac{2}{3} $$.

There is a spinner with 12 equal areas, numbered 1 through 12. If the spinner is spun one time,
what is the probability that the result is a multiple of 2 or a multiple of 3 ?
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