Paul.d or anyone else good at sums - Probability question

Discussion in 'Bulletin Board ARCHIVE' started by Farnham_Red, May 8, 2008.

  1. Farnham_Red

    Farnham_Red Administrator Staff Member Admin

    Joined:
    Jul 18, 2005
    Messages:
    34,499
    Likes Received:
    23,941
    Trophy Points:
    113
    Location:
    Farnham
    Style:
    Barnsley
    maybe you can put me out of my misery.

    I've forgotton how to do probability calculations

    I saw Napoleon Dynamites post

    http://www.barnsleyfc.org.uk/bbs/forums/thread-view.asp?tid=109496&mid=712557#M712557

    And thought for a bit of fun I would work out the chances

    Started with a cheat


    From the National Lottery's own estimation - odds of winning with a single ticket are

    Winning Selection Odds Est. prize

    Jackpot, Match 6 main numbers 1 in 13,983,816 £2,000,000
    Match 5 main numbers + the bonus 1 in 2,330,636 £100,000
    Match 5 main numbers 1 in 55,492 £1,500
    Match 4 main numbers 1 in 1,033 £60
    Match 3 main numbers 1 in 57 £10

    source
    http://www.lottery.co.uk/lotto.asp

    So to win with any single ticket with 4 or more numbers I think the odds are 1 in 1014

    (1/1033 + 1/55492 + 1/2million + 1/14million = 1/1014)

    This means the chances of not winning if you buy 50 tickets is 1013^50 / 1014^50 = 0.95

    ie the chances of winning in one go is 0.05 or 1 in 20


    So far so good

    But then I realised that he could also get 5 or more £10 wins in the 50 lines, and I cant figure out how to work it out - anyone care to remind me
     
    Farnham_Red, May 8, 2008
    #1
  2. bfc

    bfchris Member

    Joined:
    Aug 1, 2005
    Messages:
    958
    Likes Received:
    7
    Trophy Points:
    18
    Location:
    Barnsley, United Kingdom
    Home Page:
    Style:
    Barnsley (full width)
    Are you assuming....

    that all lines give you a different set of numbers and that none of the numbers are duplicated on each of the 50 lines? Does this matter or am I just overcomlicating things? I started trying to work it out but got slightly different probabiliies due to the bonus ball!</p>

    Jackpot (all six winning numbers selected) There are a total of 13,983,816 different groups of six numbers which could be drawn from the set {1, 2, ... , 49}. To see this we observe that there are 49 possibilities for the first number drawn, following which there are 48 possibilities for the second number, 47 for the third, 46 for the fourth, 45 for the fifth, and 44 for the sixth. If we multiply the numbers 49 x 48 x 47 x 46 x 45 x 44 we get 10,068,347,520. However, each possible group of six numbers (combination) can be drawn in different ways depending on which number in the group was drawn first, which was drawn second, and so on. There are 6 choices for the first, 5 for the second, 4 for the third, 3 for the fourth, 2 for the fifth, and 1 for the sixth. Multiply these numbers out to arrive at 6 x 5 x 4 x 3 x 2 x 1 = 720. We then need to divide 10,068,347,520 by 720 to arrive at the figure 13,983,816 as the number of different groups of six numbers (different picks). Since all numbers are assumed to be equally likely and since the probability of some number being drawn must be one, it follows that each pick of six numbers has a probability of 1/13,983,816 = 0.00000007151. </p>

    Five plus bonus: The pick of six must include 5 winning numbers plus the bonus. Since 5 of the six winning numbers must be picked, this means that one of the winning numbers must be excluded. There are six possibilities for the choice of excluded number and hence there are six ways for a pick of six to win the second place prize. The probability is thus 6/13,983,816 = 0.0000004291 which translates into odds against of 2,330,635:1. </p>

    Do you not just have 50 times more chance of winning? i. e 50 x six balls plus 50 x 5 balls etc?</p>

    My head hurts!!!!!</p>
     
    bfchris, May 8, 2008
    #2
  3. pau

    paul.d Well-Known Member

    Joined:
    Jul 18, 2005
    Messages:
    12,904
    Likes Received:
    754
    Trophy Points:
    113
    Gender:
    Male
    Style:
    Barnsley (full width)
    paul.d, May 8, 2008
    #3
  4. Farnham_Red

    Farnham_Red Administrator Staff Member Admin

    Joined:
    Jul 18, 2005
    Messages:
    34,499
    Likes Received:
    23,941
    Trophy Points:
    113
    Location:
    Farnham
    Style:
    Barnsley
    My head was hurting too

    but you cant just multiply

    ie the chances of winning £10 is 1/57 so by your logic if you had 57 goes you would be certain to win

    wheras the actual chances of winning are £10 if you have 57 goes is actually 0.635 or 36/57

    ie the chances of not winning are 56/57 x 56/57 x56/57.... (57 times) = 0.365. so the chances of winning are 1- that
     
    Farnham_Red, May 8, 2008
    #4

Share This Page