Possum (possum_31415) wrote,
Possum
possum_31415

This is one of the questions from a series of probability questions set by a user called 'getting better' on the Betfair forum.

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|  General Betting   :   Probability Quiz No. 15 - Ginger Hunt
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Posted by : getting better 22 Jul 13:23

My mother needs your help to working out probability of being able to buy some crystallized ginger to make a cake. It is very hard to get in the local town; sometimes this is because the crop has failed, and there is none available anywhere, but usually it is because of the unpredictability of the stock control systems of the local shops, which operate independently of each other.

The best chance is Tesco, where the probability of being able to buy ginger is the same as that of the crop having failed. The least likely is Somerfield, where there is only half the chance as at Tesco.

The probability that at least one of the shops on this side of town, Asda or Somerfield, has the ginger is the same as that of Tesco having it.

If the crop hasn’t failed, then there is a one third chance of being able to buy ginger on the other side of the town, where Tesco and Waitrose are based.

If the sum of the probabilities of being able to buy ginger at each of the 4 shops is 0.7125, what is the chance of being able to buy crystallized ginger somewhere in town?

Posted by : Jd777 22 Jul 13:28

hate to say it but still getting no solution gb

Posted by : galejo 22 Jul 14:14

Agree with JD, Waitrose comes out as -4.65% I think.

Posted by : BotMan 22 Jul 14:20

agreed, Waitrose negative

Posted by : Possum 22 Jul 14:50

> The best chance is Tesco, where the probability of being able to buy > ginger is the same as that of the crop having failed.

Is that really what you mean? Not "of the crop having succeeded"?

Posted by : getting better 22 Jul 15:00

just back from lunch, I'll have a look

Posted by : getting better 22 Jul 15:26

My mother needs your help to working out probability of being able to buy some crystallized ginger to make a cake. It is very hard to get in the local town; sometimes this is because the crop has failed, and there is none available anywhere, but usually it is because of the unpredictability of the stock control systems of the local shops, which operate independently of each other.

The best chance is Tesco, where the probability of being able to buy ginger is the same as that of the crop having failed. The least likely is Somerfield, where there is only half the chance as at Tesco.

The probability that at least one of the shops on this side of town, Asda or Somerfield, has the ginger is the same as that of Tesco having it.

If the crop hasn’t failed, then there is an evens chance of being able to buy ginger on the other side of the town, where Tesco and Waitrose are based.

If the sum of the probabilities of being able to buy ginger at each of the 4 shops is 0.7125, what is the chance of being able to buy crystallized ginger somewhere in town?

Posted by : getting better 22 Jul 15:26

The original question was OK, JDS confused me

Posted by : BotMan 22 Jul 15:29

41.667% or 5/12

Posted by : galejo 22 Jul 15:31

Well Tesco is no longer the best chance, but:

Prob of crop not failing = 85.38% Prob of buying anywhere given crop hasn't failed = 57.31%

or Prob of buying anywhere full stop is 85.38% * 57.31% = 48.93%

Posted by : galejo 22 Jul 15:33

Gave two answers 'cause I wasn't sure whether you wanted prob of buying anywhere given crop hasn't failed or without knowing whether crop had failed or not.

Posted by : Jd777 22 Jul 15:49

hmm sorry about that gb, I will recheck!

Posted by : Jd777 22 Jul 16:06

Sorry gb, here were my workings tho - pehaps someone can spot my mistake let x be the prob of crop failure and t,s,w,a be probs of supermarkets having stock t=x and s = x/2 Prob(A or S) = t, so a + s - as = t, and so a(1-x/2) = x/2 and so a = x/(2-x) Prob(T or W | F') = 0.5. , Prob(T or W | F') = Prob ((T or W) and F')/Prob(F') and as the supermarkets can only be stocked when the crop hasn't failed Prob ((T or W) and F') = Prob(T or W).

so t + w - tw = 0.5(1-f) so w(1-x) = 0.5(1-x) - x so w = (1-3x) / 2(1-x)

hence my equation from b4: 3x/2 + (1-3x)/2(1-x) + x/(2-x) = 0.7125, which has no solution x in [0,1]

Posted by : Mac 22 Jul 16:13

After all that... 50% exactly!

Posted by : Mac 22 Jul 16:15

Jd:

Prob(A or S) = t, so a + s - as = t, and so a(1-x/2) = x/2 and so a = x/(2-x)

A and S are not independent...

Posted by : Jd777 22 Jul 16:16

it says they are in the question

Posted by : BotMan 22 Jul 16:21

Jd: you have to consider the cases of crop failing and crop succeeding separately when combining probabilities of two shops: e.g. if crop succeeds a(1-x)+s(1-x)-a(1-x)s(1-x)=t(1-x)

Posted by : albatross 22 Jul 16:22

51.49%

Posted by : galejo 22 Jul 16:32

Reading it again, there's a lot of ambiguity about the conditional parts to this question.

1. The prob that Tesco has stock is the same as that of the crop failing. This has to refer to the conditional prob that Tesco has stock given that the crop has not failed, otherwise the 50% chance of there being stock at Tesco or Waitrose given the crop has not failed is clearly wrong (it would be 100%).

2. If the Tesco probability from above is conditional then presumably the Safeway (half of Tesco) is also conditional.

3. The Waitrose or Tesco probability of 50% is clearly stated as conditional, but the big question is, is the Asda or Safeway probability (equal to Tesco) also conditional - I took it to be so as all the others were also and as a further step also took the sum of the 4 shops quoted (equal to 71.25%) to also be conditional of the crop having failed.

I think some clarification on the conditional parts could be needed ...

Posted by : galejo 22 Jul 16:38

Scrap that - sounds rubbish reading it back.

Posted by : Jd777 22 Jul 16:39

Can't agree Gal.

The question as stated implies that none of the probabilities are conditional other than the one that is explicitly stated as so

Posted by : BotMan 22 Jul 16:44

"If the crop hasn't failed" is written as applying only to the Tesco/Waitrose statement. I've taken all the other statements as unconditional.

Posted by : galejo 22 Jul 16:46

You're right jd - if none of them are conditional save the one stated then Waitrose remains -ve (-4.58% I think).

Only way I could solve it was assuming the conditional bit, but there is no reason to assume that as you state.

Posted by : getting better 22 Jul 17:04

As JD says none of the probabilties are conditional except where stated, but I really can't see why there isn't a positive solution (there may be negaitve or non-real solutions as well but my mother won't like them!)

Posted by : getting better 22 Jul 17:17

but read the probability of two stores being out of stock due to a crop failure is not indepenent of cOurse!

Posted by : BotMan 22 Jul 17:23

confirms 5/12 and if crop doesn't fail stock probabilities are 1/3, 1/4, 1/5, 1/6

Posted by : getting better 22 Jul 17:32

Possum in response to the orignial question thta is indeed what I mean sorry about the delay confirming that. There seem to be two camps forming, those who belvie the questionn has an answer and those who don't!

Posted by : BotMan 22 Jul 17:39

oops not 5/12, 50%, Mac is right, answer is 1-(1/4 + (1/3)*(3/4)) forgot the 3/4!

Posted by : Jd777 22 Jul 17:41

BotMan - I agree with what you have said in principle, but still get negative answers.

let x be Pr(failure) and t,s,a,w be probs CONDITIONAL on crop success. These are now independent.

t(1-x) = x ie. t = x/(1-x) and also s = x/2(1-x)

Prob (asda or sains have stock) = (1-x)(a + s -as) = t(1-x), so a + s -as = t Prob (tesco or wait have stock given success) = t + w - tw = 0.5 Sum of probs = (1-x)(t+s+a+w) = 0.7125

I still get a negative for waitrose

Posted by : Jd777 22 Jul 17:50

guess that puts me in agreement with gal ie get waitrose solving to -4.6% roughly

Posted by : BotMan 22 Jul 18:00

Sorry Jd, I think my algebra was a bit scrambled in my earlier reply

If c is probability of a crop and (1-c) of failure If there is a crop, stocking probabilites are t, s, a, w So overall probabilities are ct, cs, ca, cw

then ct=1-c s=t/2 a+s-as=t t+w-tw=1/2 ct+cs+ca+cw=0.7125

solve by substitution or by putting in an arbitrary value for c and adjusting it until the last statement agrees

then probability of success is 1-((1-c)+c(1-t)(1-s)(1-a)(1-s)) and the answer is 0.5

it's Somerfield, by the way, gb's mother never goes to Sainsbury's

Posted by : Mac 22 Jul 19:06

Probability of crop failure is 1/4, if that helps with calculations etc

Posted by : Jd777 22 Jul 20:13

ok I agree now - 50%

my algebra is fine but my numerical method must have converged on an improper root of the equation!

Posted by : Mac 22 Jul 20:42

Yes - there is a solution where the Tesco, Waitrose, Somerfield and Asda probs are 0.34761, -0.04584, 0.173805 and 0.236925 respectively. Interestingly if you ignore the negative figure and just calculate the answer you still get 50%

Posted by : Jd777 22 Jul 21:27

hmm suggests that there's a shortcut without all this algebra...

Posted by : Possum 23 Jul 01:29

solving the cubic equation 520c^3 - 858c^2 + 435c - 63 = 0 gives three different possible values for c:

c = 0.247610434 leads to t, s, a, w = 0.752389566 0.376194783 -0.724425289 0.308340884 c = 0.652389575 leads to t, s, a, w = 0.347610425 0.173805213 0.236925227 -0.045840862 c = 0.750000000 leads to t, s, a, w = 0.250000002 0.125000001 0.150000001 0.187499997

all 3 solutions lead to a final answer of 0.50. odd, isn't it?

Anyway, that's my answer - 0.50.

Posted by : getting better 23 Jul 09:06

In jd's notation: Answer= c- c(1-a).(1-s).(1-w).(1-t) =c-c(1-t).(1-.5) =c/2+ct/2 =c/2+(1-c)/2 =0.5

Posted by : Jd777 23 Jul 09:21

like it gb

Posted by : Possum 23 Jul 11:22

how terribly clever!

gb - you often give the impression of not knowing the answer when you set these questions, so how come they work out so nicely?

this time, for instance, you weren't even sure that there WAS an answer, and yet you still end up with a cute little solution.

it's a kind of magic!

Posted by : BotMan 23 Jul 11:53

agree with Possum - a real beauty - looked like an ugly bastard, I mean .7125 - I ask you, then it falls out to really simple numbers: brilliant!

Posted by : Mac 23 Jul 12:43

Very nice Not clear how you go from c- c(1-a).(1-s).(1-w).(1-t) to c - c(1-t)(1-0.5) and/or how you use the 0.7125 but very nice all the same.

Posted by : getting better 23 Jul 12:54

In the question it says that:(1-a)(1-s)=(1-t) as this is the chance of not being able to get ginger on this side of the town, and the chance of not being able to get it at Tesco. It also says that if the crop has not failed there is a 0.5 chance of getting it on the far side of town, i.e. (1-w)(1-t)=0.5. You don't need the 0.7125 to get the answer, but the extra information actually makes the question harder because it distracts you from the simpler routes to the answer!.

Posted by : Mac 23 Jul 13:12

Quite delightful - hats off to you gb!

Posted by : getting better 24 Jul 06:43

Well done mac - 10 points

I think the others should be botman 3, jd777 2 and possum 3 (botman and jd777 both wrong on first answers)

Thanks for the kind comments, I'll try to keep them coming

Actually there may already be one on the pipeline as my colleague with the kids who went to chitty chitty bang bang has entered his children for a tennis competition which has some very complicated handicapping rules, but he is re-checking with organisers what the exact rules are.

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