Two letters are chosen randomly from the word PROBABILITY. Find the probability that the two letters chosen are the same.

The word PROBABILITY has 11 letters in total.

First, let's break down the frequency of each letter:

• P: 1
• R: 1
• O: 1
• B: 2
• A: 1
• I: 2
• L: 1
• T: 1
• Y: 1

The only letters that appear more than once are B and I.

The probability of drawing two of the same letters depends on these two letters.

1. The probability of drawing two Bs:

   • The first B has a 2/11 chance of being drawn.
   • After one B is drawn, there is only 1 B left, and 10 total letters. So, the second B has a 1/10 chance of being drawn.
   • Therefore, the probability of drawing two Bs is (2/11) * (1/10) = 2/110 = 1/55.

2. The probability of drawing two Is:

   • The first I has a 2/11 chance of being drawn.
   • After one I is drawn, there is only 1 I left, and 10 total letters. So, the second I has a 1/10 chance of being drawn.
   • Therefore, the probability of drawing two Is is (2/11) * (1/10) = 2/110 = 1/55.

Since these are mutually exclusive events (you can't draw two Bs and two Is at the same time), you can add the probabilities together to get the total probability of drawing two of the same letter.

So, the total probability is 1/55 + 1/55 = 2/55.