Question

you spin the spinner twice.
what is the probability of landing on an odd number and then landing on a number less than 8?
write your answer as a fraction or whole number.

Explanation:

Step1: Determine total sections

The spinner has 3 sections (6, 7, 8), so total outcomes for one spin: \( n = 3 \).

Step2: Probability of odd number

Odd numbers on spinner: 7 (only 7 is odd). So \( P(\text{odd}) = \frac{1}{3} \).

Step3: Probability of number < 8

Numbers less than 8: 6, 7 (two numbers). So \( P(\text{<8}) = \frac{2}{3} \).

Step4: Multiply probabilities (independent events)

Since spins are independent, \( P(\text{odd then <8}) = P(\text{odd}) \times P(\text{<8}) = \frac{1}{3} \times \frac{2}{3} = \frac{2}{9} \). Wait, no—wait, wait, let's recheck. Wait, first spin: odd number. The numbers are 6 (even), 7 (odd), 8 (even). So odd is 1 out of 3. Then number less than 8: 6 and 7, so 2 out of 3. So multiplying: \( \frac{1}{3} \times \frac{2}{3} = \frac{2}{9} \)? Wait, no, wait—wait, maybe I miscounted. Wait the spinner: 6 (purple), 7 (green), 8 (blue). So three sections. So first, odd number: only 7, so 1/3. Then number less than 8: 6 and 7, so 2/3. So the combined probability is (1/3)(2/3) = 2/9? Wait, no, wait—wait, maybe the spinner sections are equal? The problem shows a spinner with three sections: 6,7,8. So each section is equally likely? So yes, each has probability 1/3. So first spin: odd (7) → 1/3. Second spin: number less than 8 (6 or 7) → 2/3. So multiply: (1/3)(2/3) = 2/9? Wait, no, wait—wait, maybe I made a mistake. Wait, let's re-express:

Wait, first event: landing on odd number. The numbers are 6 (even), 7 (odd), 8 (even). So odd is 7, so 1 favorable out of 3. So \( P(\text{odd}) = \frac{1}{3} \).

Second event: landing on number less than 8. Numbers less than 8: 6, 7. So 2 favorable out of 3. So \( P(\text{<8}) = \frac{2}{3} \).

Since the two spins are independent, the probability of both events is the product: \( \frac{1}{3} \times \frac{2}{3} = \frac{2}{9} \). Wait, but let's check again. Wait, maybe the spinner sections are not equal? But the diagram shows three sections: 6,7,8. Assuming equal probability (since it's a spinner with three regions, probably equal). So yes, each has probability 1/3. So the calculation is correct. Wait, but wait—wait, maybe I messed up the second event. Number less than 8: 6 and 7, so two numbers, so 2/3. First event: odd, which is 7, so 1/3. So multiplying gives 2/9. Wait, but let's confirm.

Wait, another way: list all possible two-spin outcomes. First spin: 6,7,8. Second spin: 6,7,8. So total outcomes: 3*3=9.

Now, outcomes where first is odd (7) and second is <8 (6 or 7):

First spin: 7, second spin: 6 → (7,6)

First spin: 7, second spin: 7 → (7,7)

So two outcomes. So total favorable: 2. Total possible: 9. So probability is 2/9. Yes, that matches. So the correct probability is 2/9.

Answer:

\(\frac{2}{9}\)