One letter is chosen at random from the word ASSISTANT and another letter is chosen at random from the word STATISTICS. What is the probability that the two chosen letters are the same?

Introduction / Context:
This question tests probability with unequal letter frequencies in two different words. The idea is to select one letter from each word and find the probability that both letters match. Because letters occur with different frequencies, we must account for these frequencies when calculating probabilities.

Given Data / Assumptions:

• Word 1: ASSISTANT (9 letters).
• Word 2: STATISTICS (10 letters).
• One letter is chosen uniformly at random from each word.
• We want the letters to be identical.

Concept / Approach:
The probability that both letters are the same is the sum over all letters that appear in both words of [probability of choosing that letter from ASSISTANT] * [probability of choosing that letter from STATISTICS]. We must first count how many times each relevant letter occurs in each word, then use these counts over 9 and 10 respectively.

Step-by-Step Solution:
In ASSISTANT: A = 2, S = 3, I = 1, T = 2, N = 1, total = 9. In STATISTICS: S = 3, T = 3, A = 1, I = 2, C = 1, total = 10. Relevant common letters: A, S, I, T. P(both A) = (2/9) * (1/10) = 2/90. P(both S) = (3/9) * (3/10) = 9/90. P(both I) = (1/9) * (2/10) = 2/90. P(both T) = (2/9) * (3/10) = 6/90. Total probability = (2 + 9 + 2 + 6) / 90 = 19 / 90.

Verification / Alternative check:
We can compute each letter contribution with fractions directly: 2/9 * 1/10, 3/9 * 3/10, 1/9 * 2/10, 2/9 * 3/10. Adding them with a common denominator 90 again gives 19/90. Because the denominator already reflects independent choices, there is no further simplification needed beyond 19/90.

Why Other Options Are Wrong:
35/96 and 19/96 use an incorrect denominator and do not match the detailed count. None of these is incorrect because we have found an exact match 19/90 in the options.

Common Pitfalls:
A common error is to assume each distinct letter has equal probability, rather than weighting by frequency. Another mistake is to add probabilities of selecting letters without multiplying for the two independent choices. Always multiply probabilities when two selections are independent and then sum over possible matching letters.

Final Answer:
Hence, the probability that the two selected letters are the same is 19/90.