Results 1 to 4 of 4

Math Help - proportion of numbers that contain a 3

  1. #1
    Member
    Joined
    Dec 2009
    Posts
    225

    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?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Quote Originally Posted by demode View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Quote Originally Posted by demode View Post

    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.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Quote Originally Posted by demode View Post
    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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Inverse proportion/Direct proportion question
    Posted in the Statistics Forum
    Replies: 3
    Last Post: September 26th 2011, 08:22 AM
  2. proportion
    Posted in the Algebra Forum
    Replies: 1
    Last Post: September 14th 2011, 05:31 PM
  3. proportion
    Posted in the Algebra Forum
    Replies: 1
    Last Post: September 13th 2011, 05:08 PM
  4. Replies: 3
    Last Post: July 6th 2009, 08:25 PM
  5. Proportion again !!!
    Posted in the Algebra Forum
    Replies: 2
    Last Post: June 15th 2008, 01:59 AM

Search Tags


/mathhelpforum @mathhelpforum