Question

a spinner contains four equally shaded areas, as shown below. louise spins the spinner twice. which probabilities are correct? check all that apply. \\(\square\\) \\(p(\text{both green}) = \frac{1}{16}\\) \\(\square\\) \\(p(\text{both blue}) = \frac{1}{4}\\) \\(\square\\) \\(p(\text{first green and then blue}) = \frac{1}{6}\\) \\(\square\\) \\(p(\text{first orange and then blue}) = \frac{1}{8}\\) \\(\square\\) \\(p(\text{first orange and then green}) = \frac{1}{4}\\)

Explanation:

First, let's determine the probability of each color on a single spin. The spinner has 4 equally shaded areas: 1 orange, 2 blue, and 1 green. So:

• \( P(\text{orange}) = \frac{1}{4} \)
• \( P(\text{blue}) = \frac{2}{4} = \frac{1}{2} \)
• \( P(\text{green}) = \frac{1}{4} \)

Since the spins are independent, we can multiply the probabilities for each event.

Step 1: Check \( P(\text{both green}) \)

\( P(\text{green}) \times P(\text{green}) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16} \). This is correct.

Step 2: Check \( P(\text{both blue}) \)

\( P(\text{blue}) \times P(\text{blue}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \). This is correct.

Step 3: Check \( P(\text{first green and then blue}) \)

\( P(\text{green}) \times P(\text{blue}) = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}
eq \frac{1}{6} \). This is incorrect.

Step 4: Check \( P(\text{first orange and then blue}) \)

\( P(\text{orange}) \times P(\text{blue}) = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \). This is correct.

Step 5: Check \( P(\text{first orange and then green}) \)

\( P(\text{orange}) \times P(\text{green}) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}
eq \frac{1}{4} \). This is incorrect.

Answer:

• \( P(\text{both green}) = \frac{1}{16} \) (correct)
• \( P(\text{both blue}) = \frac{1}{4} \) (correct)
• \( P(\text{first orange and then blue}) = \frac{1}{8} \) (correct)