A spinner is divided into 3 equal sections, as shown. An arrow is attached to the centre of the spinner. The arrow is spun twice. What is the probability that the arrow lands on the same colour twice?
(A) $\frac{1}{9}$
(B) $\frac{2}{3}$
(D) $\frac{1}{3}$
(E) $\frac{2}{9}$
(C) $\frac{1}{2}$

![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-020.jpg?height=344&width=347&top_left_y=1384&top_left_x=1320)
Solution 1

After the arrow is spun twice, the possible outcomes are: red, red; red, blue; red, green; blue, red; blue, blue; blue, green, green, red; green, blue; green, green.

That is, there are 9 possible outcomes.

Of these 9 outcomes, 3 have the same colour appearing twice: red, red; blue, blue; green, green. The probability that the arrow lands on the same colour twice is $\frac{3}{9}=\frac{1}{3}$.

## Solution 2

First, we count the total number of possible outcomes given that the arrow is spun twice.

When the arrow is spun the first time, there are 3 possible outcomes (red, blue, green).

When the arrow is spun the second time, there are again 3 possible outcomes.

Thus when the arrow is spun twice, there are $3 \times 3=9$ possible outcomes.

Next, we count the number of outcomes in which the arrow lands on the same colour twice.

When the arrow is spun the first time, there are 3 possible outcomes (red, blue, green).

When the arrow is spun the second time, there is 1 possible outcome (since the colour must match the colour that the arrow landed on after the first spin).

Thus when the arrow is spun twice, there are $3 \times 1=3$ possible outcomes in which the arrow lands on the same colour twice.

The probability that the arrow lands on the same colour twice is $\frac{3}{9}=\frac{1}{3}$.

## Solution 3

The first spin will land on one of the three colours.

For the arrow to land on the same colour twice, the second spin must land on the colour matching the first spin.

Since exactly 1 of the 3 colours matches the first spin, the probability that the arrow lands on the same colour twice is $\frac{1}{3}$.

ANswer: (D)