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# Counting

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I need help with this

Sam writes down the numbers $1,$ $2,$ $\dots,$ $315,$ $316,$ $317,$ $\dots,$ $249,$ $250.$

(a) How many digits did Sam write, in total?

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

Oct 28, 2023

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(a)

To count the number of digits that Sam wrote, we can use the following formula:

Number of digits = Ceiling(log10(Largest number))

The largest number that Sam wrote is 250, so the number of digits that he wrote is:

Number of digits = Ceiling(log10(250)) = Ceiling(2.3979) = \boxed{3}

(b)

To find the probability that Sam chooses a 2, we can use the following formula:

Probability = (Number of favorable outcomes) / (Total number of outcomes)

The number of favorable outcomes is the number of times the digit 2 appears in the numbers that Sam wrote down. The number 2 appears 50 times, since there are 50 even numbers from 1 to 250.

The total number of outcomes is the total number of digits that Sam wrote down, which is 3.

Therefore, the probability that Sam chooses a 2 is:

Probability = 50 / 3 = \boxed{\frac{50}{3}}

Python code

The following Python code can be used to calculate the number of digits that Sam wrote and the probability that he chooses a 2:

Python

def count_digits(n): """Counts the number of digits in a positive integer.""" count = 0 while n > 0: count += 1 n //= 10 return count def count_2s(n): """Counts the number of 2s in a positive integer.""" count = 0 while n > 0: if n % 10 == 2: count += 1 n //= 10 return count # Count the number of digits in the numbers from 1 to 250. total_digits = 0 for i in range(1, 251): total_digits += count_digits(i) # Count the number of 2s in the numbers from 1 to 250. total_2s = count_2s(250) # Calculate the probability of choosing a 2. probability = total_2s / total_digits print(f"The number of digits is {total_digits}.") print(f"The probability of choosing a 2 is {probability}.")