Thread: proportion of numbers that contain a 3

1. proportion of numbers that contain a 3

1. How many natural numbers less than 100 contain a 3? (Note that 13, 35 and 73 all contain a 3 but 42, 65 and 88 do not). What proportion is this?

2. Approximately what proportion of numbers less than 1000 contain a 3?

3. Explain why almost all million-digit numbers contain at least one 3.

P.S. I know that there are 19 numbers less than 100 that contain 3 but I found that by counting to 100. Then how do I find the proportions?

2. Originally Posted by demode
1. How many natural numbers less than 100 contain a 3? (Note that 13, 35 and 73 all contain a 3 but 42, 65 and 88 do not). What proportion is this?

2. Approximately what proportion of numbers less than 1000 contain a 3?

3. Explain why almost all million-digit numbers contain at least one 3.

P.S. I know that there are 19 numbers less than 100 that contain 3 but I found that by counting to 100. Then how do I find the proportions?
Q1.

Ten numbers from 10 to 99 start with 3..... 30, 31, 32, 33, 34...
one number from 0 to 9 is 3,
nine numbers from 10 to 99 end with 3..... 13, 23, 33, 43....

33 is double counted.

Alternatively,
3 occurs in the 10's position 10 times......30, 31 to 39

and 3 occurs in the units position 10 times.....3, 13, 23, 33, 43...

33 is in both groups.

3. Originally Posted by demode

2. Approximately what proportion of numbers less than 1000 contain a 3?

P.S. I know that there are 19 numbers less than 100 that contain 3 but I found that by counting to 100. Then how do I find the proportions?
If you take 0 as a natural number, then the proportion for Q1 is $\frac{19}{100}$

Q2.

3 can appear in the units, tens and hundreds positions.
You can avoid double-counting as follows

03XX occurs $10^2$ times, since X can be 3.

0Y3X occurs $9(10)$ times, where Y cannot equal 3 to avoid double-counting.

Y can be 0, 1, 2, 4, 5, 6, 7, 8, 9.

0YY3 occurs $9^2$ times

There are $10^3$ numbers from 0 to 999.

Then you can express the proportion as a fraction.

4. Originally Posted by demode
1. How many natural numbers less than 100 contain a 3? (Note that 13, 35 and 73 all contain a 3 but 42, 65 and 88 do not). What proportion is this?

2. Approximately what proportion of numbers less than 1000 contain a 3?

3. Explain why almost all million-digit numbers contain at least one 3.

P.S. I know that there are 19 numbers less than 100 that contain 3 but I found that by counting to 100. Then how do I find the proportions?
Quite possibly, due to the term "approximation",
these questions may be in the context of calculating the probability of having a 3.

Hence, for Q3...

the probability of a million digit number not containing a 3 is

$\frac{8}{9}\left(\frac{9}{10}\right)^{999,999}$

which is pretty small.

Then the probability of a million digit number containing at least one 3 is

$1-\frac{8}{9}\left(0.9\right)^{999,999}$

which is practically indistinguishable from 1.

,

,

what proportion of the first 10000 natural

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