To find the probability of landing on a divisor of 24 and then landing on a prime number when spinning the spinner labeled 1, 2, and 3, we first need to determine the relevant divisors of 24 and the prime numbers from the spinner.
Step 1: Identify the divisors of 24.
The divisors of 24 are:
- 1 (since \( 1 \times 24 = 24 \))
- 2 (since \( 2 \times 12 = 24 \))
- 3 (since \( 3 \times 8 = 24 \))
- 4 (since \( 4 \times 6 = 24 \))
- 6 (since \( 6 \times 4 = 24 \))
- 8 (since \( 8 \times 3 = 24 \))
- 12 (since \( 12 \times 2 = 24 \))
- 24 (note: 24 is not on the spinner)
From the above, the divisors of 24 that are also on the spinner (1, 2, 3) are:
- 1
- 2
- 3
So, the divisors of 24 on the spinner are 1, 2, and 3, giving us a total of 3 divisors.
Step 2: Identify prime numbers from the spinner.
The numbers on the spinner are 1, 2, and 3. The prime numbers among these are:
- 2 (the only even prime)
- 3 (the next prime after 2)
Thus, the prime numbers on the spinner are 2 and 3, giving us a total of 2 prime numbers.
Step 3: Calculate the probabilities.
- Probability of landing on a divisor of 24:
The total possible outcomes when spinning the spinner once is 3 (1, 2, and 3). The number of favorable outcomes (divisors of 24) is 3 (1, 2, 3).
Thus, the probability of landing on a divisor of 24 is: \[ P(\text{Divisor of 24}) = \frac{3}{3} = 1 \]
- Probability of landing on a prime number:
Again, the total possible outcomes when spinning the spinner once is still 3. The number of favorable outcomes (prime numbers) is 2 (2 and 3).
Thus, the probability of landing on a prime number is: \[ P(\text{Prime}) = \frac{2}{3} \]
Step 4: Calculate the combined probability of both events.
Since the spins are independent, we multiply the probabilities: \[ P(\text{Divisor of 24 and then Prime}) = P(\text{Divisor of 24}) \times P(\text{Prime}) = 1 \times \frac{2}{3} = \frac{2}{3} \]
Step 5: Convert the probability to a percentage.
To convert \(\frac{2}{3}\) to a percentage: \[ \frac{2}{3} \approx 0.6667 \implies 0.6667 \times 100 \approx 66.67% \]
Rounded to the nearest tenth, the probability is: \[ \boxed{66.7%} \]