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# 1-315, how many 2's

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Sam writes down the numbers    1-315

Sam chooses one of the digits written down at random. What is the probability that Sam chooses a 2?

Apr 27, 2024

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To find the probability that Sam chooses a 2, we need to determine how many times the digit 2 appears in the numbers from 1 to 315.

First, let's count the number of times the digit 2 appears in the units place. From 1 to 315, there are 31 numbers ending in 2 (2, 12, 22, ..., 312).

Next, let's count the number of times the digit 2 appears in the tens place. From 1 to 315, there are 10 numbers in each ten block that contain the digit 2, except for the numbers from 20 to 29 where there are 11 numbers. So, there are $$10 \times 3 + 11 = 31$$ numbers with 2 in the tens place.

Finally, let's count the number of times the digit 2 appears in the hundreds place. From 1 to 315, there is only one number with 2 in the hundreds place, which is 200.

Adding up all the occurrences of the digit 2:

$31 + 31 + 1 = 63$

So, there are 63 occurrences of the digit 2 in the numbers from 1 to 315.

Now, since Sam chooses one of the digits written down at random, the total number of possible outcomes is the total number of digits written down, which is 315.

Therefore, the probability that Sam chooses a 2 is:

$P(\text{choosing a 2}) = \frac{\text{Number of occurrences of 2}}{\text{Total number of digits}} = \frac{63}{315} = \frac{1}{5}$

So, the probability that Sam chooses a 2 is $$\frac{1}{5}$$ or 20%.

Apr 28, 2024