Suppose the spinner lands on <em>a</em>. There's a 1/3 chance that it'll land on <em>a</em> the second time. Suppose the spinner lands on <em>b</em>. There's a 1/3 chance that it'll land on <em>b</em> the second time. Suppose the spinner lands on <em>c</em>. There's a 1/3 chance that it'll land on <em>c</em> the second time.
We've covered all possibilities for the first spin, and they're all equal, so their average is 1/3.
The probability that it'll land on the same letter twice is 33.3%.
