Probability of Landing on an Even Number or a Multiple of 3 on a Spinner

Suppose you have a spinner divided into 21 equal sections, labeled 1 through 18. The question is: what is the probability of the spinner landing on an even number or a multiple of 3?

The total number of sections is 18, but the problem specifies there are 21 sections, which is incorrect. However, for the sake of clarity, we will use the given number of sections (21) and analyze the numbers from 1 to 18.

Step 1: Analyzing Even Numbers

The even numbers between 1 and 18 are: 2, 4, 6, 8, 10, 12, 14, 16, 18. There are 9 even numbers.

The probability of the spinner landing on an even number is:

[ P(text{even number}) frac{9}{21} ]

Step 2: Analyzing Multiples of 3

The multiples of 3 between 1 and 18 are: 3, 6, 9, 12, 15, 18. There are 6 multiples of 3.

The probability of the spinner landing on a multiple of 3 is:

[ P(text{multiple of 3}) frac{6}{21} ]

Step 3: Analyzing Both Conditions

Some numbers satisfy both conditions: being an even number and a multiple of 3. These numbers are: 6, 12, 18. There are 3 such numbers.

The probability of the spinner landing on a number that is both an even number and a multiple of 3 is:

[ P(text{even number and multiple of 3}) frac{3}{21} ]

Step 4: Using the Inclusion-Exclusion Principle

To find the probability of the spinner landing on an even number or a multiple of 3, we use the inclusion-exclusion principle:

[ P(text{even number} cup text{multiple of 3}) P(text{even number}) P(text{multiple of 3}) - P(text{even number and multiple of 3}) ]

Substituting the probabilities:

[ P(text{even number} cup text{multiple of 3}) frac{9}{21} frac{6}{21} - frac{3}{21} frac{12}{21} frac{2}{3} ]

Thus, the probability that the spinner lands on an even number or a multiple of 3 is ( frac{2}{3} ).

Conclusion

Given the analysis, the probability of landing on an even number or a multiple of 3 is ( frac{2}{3} ). This result is consistent with the given values and the mathematical principles used in probability theory.

In summary, the key steps involved understanding the problem, identifying the relevant numbers, and applying the inclusion-exclusion principle to compute the correct probability.