A letter is taken out at random from 'ASSISTANT' and and another letter taken out from the letters of the word 'STATISTICS'. The probability that they are identical letters , is

Options

A. `13/90`
B. `1/45`
C. `19/90`
D. None of these

Answer

To solve the problem of finding the probability that a letter taken from the word "ASSISTANT" and a letter taken from the word "STATISTICS" are identical, we can follow these steps:

### Step 1: Identify the letters in each word
- The letters in "ASSISTANT" are: A, S, S, I, S, T, A, N, T
- The letters in "STATISTICS" are: S, T, A, T, I, S, T, I, C, S

### Step 2: Count the occurrences of each letter in both words
- In "ASSISTANT":
- A: 2
- S: 3
- I: 1
- T: 2
- N: 1

- In "STATISTICS":
- S: 3
- T: 3
- A: 1
- I: 2
- C: 1

### Step 3: Identify common letters
The common letters between "ASSISTANT" and "STATISTICS" are A, S, I, and T.

### Step 4: Calculate the probability for each common letter
1. **For letter A:**
- Probability of picking A from "ASSISTANT": \( P(A) = \frac{2}{9} \)
- Probability of picking A from "STATISTICS": \( P(A) = \frac{1}{10} \)
- Combined probability: \( P(A \text{ from both}) = \frac{2}{9} \times \frac{1}{10} = \frac{2}{90} \)

2. **For letter S:**
- Probability of picking S from "ASSISTANT": \( P(S) = \frac{3}{9} = \frac{1}{3} \)
- Probability of picking S from "STATISTICS": \( P(S) = \frac{3}{10} \)
- Combined probability: \( P(S \text{ from both}) = \frac{1}{3} \times \frac{3}{10} = \frac{3}{30} = \frac{1}{10} \)

3. **For letter I:**
- Probability of picking I from "ASSISTANT": \( P(I) = \frac{1}{9} \)
- Probability of picking I from "STATISTICS": \( P(I) = \frac{2}{10} = \frac{1}{5} \)
- Combined probability: \( P(I \text{ from both}) = \frac{1}{9} \times \frac{1}{5} = \frac{1}{45} \)

4. **For letter T:**
- Probability of picking T from "ASSISTANT": \( P(T) = \frac{2}{9} \)
- Probability of picking T from "STATISTICS": \( P(T) = \frac{3}{10} \)
- Combined probability: \( P(T \text{ from both}) = \frac{2}{9} \times \frac{3}{10} = \frac{6}{90} = \frac{1}{15} \)

### Step 5: Sum the probabilities of picking identical letters
Now we add the probabilities of picking each common letter:
\[ P(\text{identical letters}) = P(A) + P(S) + P(I) + P(T) \]
\[ = \frac{2}{90} + \frac{9}{90} + \frac{2}{90} + \frac{6}{90} \]
\[ = \frac{2 + 9 + 2 + 6}{90} = \frac{19}{90} \]

### Final Answer
The probability that the letters picked from "ASSISTANT" and "STATISTICS" are identical is \( \frac{19}{90} \).

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