Four digit numbers with different digits are formed using the digits `1,2,3,4,5,6,7,8`. One number from them is picked up at random. The chance that the selected number contains the digit '1' is

Explanation

To solve the problem of finding the probability that a randomly selected four-digit number formed using the digits 1, 2, 3, 4, 5, 6, 7, and 8 contains the digit '1', we can follow these steps:

### Step 1: Calculate the total number of four-digit numbers
We can form a four-digit number using the digits 1 to 8, ensuring that all digits are different.

- For the first digit, we can choose any of the 8 digits.
- For the second digit, we can choose from the remaining 7 digits.
- For the third digit, we can choose from the remaining 6 digits.
- For the fourth digit, we can choose from the remaining 5 digits.

Thus, the total number of four-digit numbers is calculated as:
\[ \text{Total numbers} = 8 \times 7 \times 6 \times 5 \]

### Step 2: Calculate the number of four-digit numbers containing the digit '1'
To find the number of four-digit numbers that contain the digit '1', we can fix '1' in one of the four positions and then fill the remaining three positions with the other digits.

- We can fix '1' in one of the 4 positions (1st, 2nd, 3rd, or 4th).
- For the remaining three positions, we can choose from the remaining 7 digits (2, 3, 4, 5, 6, 7, 8).

The choices for the remaining three digits are:
- For the first remaining position, we can choose from 7 digits.
- For the second remaining position, we can choose from 6 digits.
- For the third remaining position, we can choose from 5 digits.

Thus, the total number of four-digit numbers containing the digit '1' is calculated as:
\[ \text{Numbers containing '1'} = 4 \times (7 \times 6 \times 5) \]

### Step 3: Calculate the probability
The probability \( P \) that a randomly selected four-digit number contains the digit '1' is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[ P(\text{contains '1'}) = \frac{\text{Numbers containing '1'}}{\text{Total numbers}} = \frac{4 \times (7 \times 6 \times 5)}{8 \times (7 \times 6 \times 5)} \]

### Step 4: Simplify the probability
We can simplify the expression:
\[ P(\text{contains '1'}) = \frac{4}{8} = \frac{1}{2} \]

### Final Answer
The probability that a randomly selected four-digit number contains the digit '1' is:
\[ \frac{1}{2} \]

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