A spinner has the numbers one through five evenly spaced. If the spinner is used three times, what is the probability that it will land on an odd number exactly once ?

Options

A. `(12)/(125)`
B. `(18)/(125)`
C. `(27)/(125)`
D. `(36)/(125)`

Explanation

To find the probability that a spinner lands on an odd number exactly once when spun three times, we can follow these steps:

### Step 1: Identify the outcomes of the spinner
The spinner has the numbers 1 through 5. The odd numbers in this range are 1, 3, and 5. Therefore, there are 3 odd outcomes and 2 even outcomes (2 and 4).

### Step 2: Calculate the probabilities of landing on odd and even numbers
- Probability of landing on an odd number (P(odd)) = Number of odd outcomes / Total outcomes = 3 / 5
- Probability of landing on an even number (P(even)) = Number of even outcomes / Total outcomes = 2 / 5

### Step 3: Determine the scenario for exactly one odd number
We want to find the probability of getting exactly one odd number in three spins. This means we will have one odd number and two even numbers.

### Step 4: Calculate the probability of one specific arrangement
The probability of one specific arrangement (e.g., odd, even, even) can be calculated as follows:
- Probability = P(odd) × P(even) × P(even)
- Probability = (3/5) × (2/5) × (2/5) = (3/5) × (4/25) = 12/125

### Step 5: Count the number of arrangements
There are different arrangements for getting one odd and two even numbers. The arrangements can be:
1. Odd, Even, Even
2. Even, Odd, Even
3. Even, Even, Odd

Thus, there are 3 different arrangements.

### Step 6: Calculate the total probability
To find the total probability of getting exactly one odd number in three spins, multiply the probability of one arrangement by the number of arrangements:
- Total Probability = Number of arrangements × Probability of one arrangement
- Total Probability = 3 × (12/125) = 36/125

### Final Answer
The probability that the spinner will land on an odd number exactly once when spun three times is **36/125**.

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