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

Options

A. `1//45`
B. `13//90`
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 another taken from "STATISTICS" are the same, we can follow these steps:

### Step 1: Count the letters in each word
- In "ASSISTANT":
- A: 2
- S: 3
- I: 1
- T: 2
- N: 1

Total letters = 9

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

Total letters = 10

### Step 2: Identify common letters
The common letters between "ASSISTANT" and "STATISTICS" are:
- A
- S
- T
- I

### Step 3: Calculate the probability for each common letter
1. **For letter 'A':**
- Probability of picking 'A' from "ASSISTANT" = Number of A's in "ASSISTANT" / Total letters in "ASSISTANT" = 2/9
- Probability of picking 'A' from "STATISTICS" = Number of A's in "STATISTICS" / Total letters in "STATISTICS" = 1/10
- Combined probability for 'A' = (2/9) * (1/10) = 2/90

2. **For letter 'S':**
- Probability of picking 'S' from "ASSISTANT" = 3/9
- Probability of picking 'S' from "STATISTICS" = 3/10
- Combined probability for 'S' = (3/9) * (3/10) = 9/90

3. **For letter 'T':**
- Probability of picking 'T' from "ASSISTANT" = 2/9
- Probability of picking 'T' from "STATISTICS" = 3/10
- Combined probability for 'T' = (2/9) * (3/10) = 6/90

4. **For letter 'I':**
- Probability of picking 'I' from "ASSISTANT" = 1/9
- Probability of picking 'I' from "STATISTICS" = 2/10
- Combined probability for 'I' = (1/9) * (2/10) = 2/90

### Step 4: Add the probabilities
Now, we add the probabilities of picking the same letters:
- Total probability = Probability of A + Probability of S + Probability of T + Probability of I
- Total probability = (2/90) + (9/90) + (6/90) + (2/90) = (2 + 9 + 6 + 2) / 90 = 19/90

### Final Answer
The probability that the letters picked from "ASSISTANT" and "STATISTICS" are the same is **19/90**.

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