Question

sam writes down the numbers $1,$ $2,$ $\dots,$ $315.$ sam chooses one of the digits written down at random. what is the probability that sam chooses a $2$?

Answer 1

The probability that Sam chooses a '2' from the numbers written down from 1 to 315 is calculated by counting the frequency of the digit '2' and dividing by the total number of digits written down. The digit '2' appears 63 times, and the total number of digits is 834. Therefore, the probability is 63/834.

Step-by-step explanation:

To calculate the probability that Sam chooses a 2 from the numbers written down from 1 to 315, we need to count how many times the digit appears in the sequence. When counting, we'll see that 2 appears as:

• A unit digit for ten times every hundred numbers (i.e., 2, 12, 22, ..., 92) plus two more times in the numbers 302 and 312.
• A tens digit for ten times every hundred numbers (i.e., 20-29, 120-129, 220-229).
• A hundred digit just once in the number 200.

So, we get the frequency of 2 as:

• Units place: 3(10) + 2 = 32 times
• Tens place: 3(10) = 30 times
• Hundreds place: 1 time

Adding them together gives us 32 + 30 + 1 = 63 times the digit 2 appears from 1 to 315.

The total number of digits written is:

• (9 single digits) + (2 digits × 90 two-digit numbers) + (3 digits × 215 three-digit numbers) = 9 + 180 + 645 = 834 digits.

So, the probability that Sam chooses a 2 is:

P(2) = Total number of 2s / Total number of digits = 63/834.

Answer 2

The probability that Sam chooses a 2 when selecting a digit randomly from the numbers 1 to 250 is
`$(17)/(321).$`

To find the probability that Sam chooses a 2 when selecting a digit randomly from the numbers 1 to 250, we need to determine the total number of digits that are equal to 2 and divide it by the total number of digits in the given range.

Step 1: Count the number of times the digit 2 appears in the numbers 1 to 250.

• 1.1 Count the number of times 2 appears as a units digit:
• The numbers ending in 2 are
  `$2, 12, 22, 32, \ldots, 242.$` These numbers form an arithmetic sequence with a common difference of 10.
• So, to find the count of numbers ending in 2, we can use the formula for the number of terms in an arithmetic sequence:
• 
  `\[ \frac{\text{Last Term} - \text{First Term}}{\text{Common Difference}} + 1 = (242 - 2)/(10) + 1 = 24.\]`
• 1.2 Count the number of times 2 appears as a tens digit:
• The numbers starting with 2 in the tens place are
  `$20, 21, 22, 23, \ldots, 29.$` These numbers also form an arithmetic sequence with a common difference of 1. So, the count of numbers starting with 2 in the tens place is:
• 
  `\[ \frac{\text{Last Term} - \text{First Term}}{\text{Common Difference}} + 1 = (29 - 20)/(1) + 1 = 10.\]`

Step 2: Calculate the total number of digits in the range 1 to 250.

To find the total number of digits in this range, we can calculate the number of digits in each number separately and then sum them up.

• 2.1 Numbers with one digit (1 to 9): There are 9 single-digit numbers.
• 2.2 Numbers with two digits (10 to 99): There are $99 - 10 + 1 = 90$ two-digit numbers. Each of them has 2 digits, so there are
  `$90 \cdot 2 = 180$` digits in this range.
• 2.3 Numbers with three digits (100 to 250): There are 250 - 100 + 1 = 151 three-digit numbers. Each of them has 3 digits, so there are
  `$151 \cdot 3 = 453$` digits in this range.

Step 3: Calculate the total number of digits in the range 1 to 250.

The total number of digits is the sum of the counts from step 2:

`\[9 + 180 + 453 = 642.\]`

Step 4: Calculate the probability.

To find the probability that Sam chooses a 2, we need to divide the count of 2's (from step 1) by the total number of digits (from step 3):

`\[ \text{Probability} = \frac{\text{Count of 2's}}{\text{Total Number of Digits}} = (24 + 10)/(642) = (34)/(642) = (17)/(321).\]`

So, the answer is
`$(17)/(321).$`

The complete question is here:

Sam writes down the numbers 1,
`$ $2,$ $\dots,$ $315,$ $316,$ $317,$ $\dots,$ $249,$ $250.$`Sam chooses one of the digits written down at random. What is the probability that Sam chooses a 2?