Lottery/Set probability - Need help with a math problem
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05-08-2017, 06:19 AM
RE: Lottery/Set probability - Need help with a math problem
Lotteries are a tax on people who can't do arithmetic. Dodgy
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05-08-2017, 11:18 PM (This post was last modified: 06-08-2017 12:10 AM by Robvalue.)
RE: Lottery/Set probability - Need help with a math problem
If I'm understanding the question correctly, the answer is quite simple, 1/5.

Given that you have picked three tickets which use four overlapping numbers and the same bonus ball, and you win a level 3 prize but not the jackpot:

Like you say, for your other two tickets to win, you need the number which differs from your first ticket to be the one which isn't part of the overlap. If we know you've already won with one of your tickets, the number you missed is equally likely to be any of the five. So the probability it's the one that differs is 1 in 5.

I could try and draw it all out using the full set theory and conditional probability if you like, but I'm pretty sure it would boil down to this, unless I've made a blunder here.

Edit: I'm mow doubting myself... gonna give this more thought.

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06-08-2017, 12:30 AM (This post was last modified: 06-08-2017 12:50 AM by Robvalue.)
RE: Lottery/Set probability - Need help with a math problem
Okay, I just did this in full and got a different answer, 40/504 = 5/63

So my intuition about the problem may have been off. I'll try and write up this method at some point so you guys can analyse it. I did it fairly quickly so I may have screwed up at some point.

Edit: forget that also, I screwed up. Now the answer is apparently extremely different: 1/(7*6*5*4)

Will review Tongue

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06-08-2017, 01:26 AM (This post was last modified: 06-08-2017 01:42 AM by Robvalue.)
RE: Lottery/Set probability - Need help with a math problem
Whether we're considering level 3 or 4 prizes, it makes no difference. Then assume we've won a level 3 or 4 prize, what's the probability we win three of the same prize with this choice:

1 2 3 4 5
1 2 3 4 6
1 2 3 4 7

There are 43 ways of winning on all 3:

1 2 3 4 Y where Y is 5...47

The number of ways of winning on 1 (at least):

Number of ways of four numbers coming up from 1...7, then any remaining number = 7*6*5*4*43 This is wrong

So probability we win on 3 given we win on 1 (no jackpot) is

43 / 7*6*5*4*43

= 1 / 7*6*5*4


For fuck's sake. This is still wrong. I'll come back to this. I suck today.

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06-08-2017, 01:53 AM
RE: Lottery/Set probability - Need help with a math problem
Let's try again...

The number of ways of winning at least 1 level 3/4 prize is:

1 2 3 4 Y = 43

plus

[3 numbers from 1..4, 1 number from 5..7, any remaining number except 1..4]

= 4 * 3 * 2 * 3 * 42 = 3024

Total number of ways = 43 + 3024 = 3067

So prob of winning 3 given you've won 1 = 43 / 3067

which is about 1.5%

....did I get it right this time?

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06-08-2017, 03:39 AM
RE: Lottery/Set probability - Need help with a math problem
Hey Rob! Still wrong.

I've got it now though. I'll post later.

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06-08-2017, 09:47 AM
RE: Lottery/Set probability - Need help with a math problem
The only problem here is that lottery results don't necessarily follow the "laws" of probability. Big Grin

I'm a creationist... I believe that man created God.
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06-08-2017, 10:06 AM
RE: Lottery/Set probability - Need help with a math problem
(05-08-2017 06:19 AM)Gwaithmir Wrote:  Lotteries are a tax on people who can't do arithmetic. Dodgy

I always call it a stupidity tax.

.......................................

The difference between prayer and masturbation - is when a guy is through masturbating - he has something to show for his efforts.
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06-08-2017, 12:04 PM
RE: Lottery/Set probability - Need help with a math problem
I'm a sack of donkey bollocks.

Number of ways to get 1 or more prize:

(4 numbers 1..4, 1 number 5...47)

(3 numbers 1..4, 2 numbers 5..7)

(3 numbers 1..4, 1 number 5..7, 1 number 8..47)

= 1 * 43 + 4 * 3 + 4 * 3 * 40

= 43 + 12 + 480

= 535

So prob(3 wins given at least one win) = 43/535

...

I'll be back later to insult myself probably, or to confirm if I actually got it this time.

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07-08-2017, 01:18 AM (This post was last modified: 07-08-2017 01:22 AM by Robvalue.)
RE: Lottery/Set probability - Need help with a math problem
This problem has been so interesting to me and has revitalised my love of mathematics! After having an embaressing dialogue with myself on this thread, I'm finally confident I have it right.

I've been trying to test this out using programming, but the problem is that the probabilities here are very low and so to get any meaningful results requires very long computational times.

So instead I've been running it with 10 balls. My results confirm what I would expect from the same method.

There are 6 ways to win with all 3 tickets: 1,2,3,4 and 1 from (5,6,7,8,9,10)

There are 48 further ways to win with one ticket:

3 from (1,2,3,4) and 2 from (5,6,7) = 4*3=12
3 from (1,2,3,4) and 1 from (5,6,7) and 1 from (8,9,10) = 4*3*3=36

So this gives a total of 54, giving an expected probability of 6/54. My program confirm this, giving good evidence that my method for the 47 balls is correct. Therefor I'm putting forward 43/535 as my final answer.

I'll go further and say that that the general probability for N>5 balls is

(N - 4) / (13N - 76)

Here's the code, using Microsoft Small Basic:
one = 0
three = 0

TextWindow.WriteLine("Number of balls?")
balls = TextWindow.ReadNumber()
TextWindow.WriteLine("Number of games?")
n = TextWindow.ReadNumber()

For games = 1 To n


For t = 1 To 5
ok=0
While ok=0
ok=1
ball[t]=math.GetRandomNumber(balls)
If t>1 Then
For x = 1 to t-1
If ball[x]=ball[t] then
ok=0
EndIf
EndFor

EndIf


EndWhile



EndFor
hit=0
near=0
For x = 1 To 5
If ball[x] < 5 Then
hit = hit+1
EndIf
If ball[x] >4 And ball[x] < 8 Then
near=near+1
EndIf

EndFor
If hit > 3 Then
three = three+1
one=one+1
EndIf
if hit = 3 And near >0 Then
one = one +1
EndIf




TextWindow.WriteLine("")
TextWindow.WriteLine("Times three tickets won:")
TextWindow.WriteLine(three)
TextWindow.WriteLine("Times at least one ticket won:")
TextWindow.Writeline(one)

If one>0 then
TextWindow.WriteLine("Percentage conditional probability:")
TextWindow.Writeline(three*100/one)
EndIf

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