+1791 Sam writes down the numbers $1,$ $2,$ $\dots,$ $315,$ $316,$ $317,$ $\dots,$ $248,$ $249,$ $250.$
(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 $2$?
+618 Absolutely, I've been improving my problem-solving abilities in counting problems. Let's tackle this step-by-step.
(a) (Observe the digits of the numbers 1 to 250).
Numbers from 1 to 9 have 1 digit each (9 numbers).
Numbers from 10 to 99 have 2 digits each (90 numbers).
Numbers from 100 to 250 have 3 digits each (151 numbers).
Therefore, the total number of digits is 9×1+90×2+151×3=9+180+453=642.
(b) (Calculate the number of times the digit 2 appears)
Each number from 10 to 19 contains the digit 2 once. (10 numbers)
Each number from 20 to 29 contains the digit 2 once. (10 numbers)
Similarly, each hundred from 100 to 240 contains the digit 2 ten times (15 hundreds * 10 = 150 times)
Additionally, numbers 200, 210, 220, 230, and 240 each contain the digit 2 twice. (5 numbers * 2 = 10 times)
So, the total number of times the digit 2 appears is 10+10+150+10=180.
(c) The probability of Sam choosing a 2 is the number of times 2 appears divided by the total number of digits:
Probability = (Number of 2s) / (Total number of digits) = 180 / 642 = 20/71
Therefore, the probability of Sam choosing a 2 is 20/71.
+128475 No. of 2's
In the ones place
10 in the first hundred
10 in the second hundred
5 in last 50
= 25
In the tens place
10 in the first hundred
10 in the second hundred
10 in the last 50
= 30
In the hundreds place
51 in the last 51
Total 2's = [ 25 + 30 + 51 ] = 106
Prob of a 2 = 106 / 642 = 53 / 321
