Million dollar question
A logician puts 10,000 dollar in a Weekly trading scheme.
-> About half the time, He makes an 80% gain.
-> The other half, he makes a 60% loss.
One year later, how much money he will have ?
a> 1.95
b> 14,000
c> 140,000
d> 1.4 Million
e> 131 Million
No Marks without explaination !
-> About half the time, He makes an 80% gain.
-> The other half, he makes a 60% loss.
One year later, how much money he will have ?
a> 1.95
b> 14,000
c> 140,000
d> 1.4 Million
e> 131 Million
No Marks without explaination !
Labels: friday special, logic, mathemagic, puzzle
posted by Rajesh Lal at
9:29 AM
29 Comments:
1.95, given that the loss is 28% every 2 weeks.
1.95
I'm assuming you meant "exactly half the time", not "about..." If the distribution of gains and losses are not equal, the answer becomes chaotic and essentially random after only 52 samples.
Constraining gains and losses means that every two weeks, the loss will be 1.8*0.4 = .72.
10,000 * (.72 ^ 26) = 1.95 (rounded)
Because the distribution is equal and multiplication is commutive, it doesn't matter what order the gains and losses are in. You always get the same result, although if you happen to get 26 gains in a row (and therefore know you'll now get 26 losses in a row), then you should split town and disappear with the $43B.
if he gain 80% and loses 60% it means there's a 20% gain weekly from the trade so he should make 96000 in a year
Think of it this way:
10,000 * (1.8 * 0.4 * 1.8 * 0.4 ...) = 1.95
You're multiplying the numbers, not averaging them:
(1.8 + 0.4) / 2 [the "average"] does not equal
(1.8 * 0.4) over two weeks.
This post has been removed by a blog administrator.
The only thing more pathetic is the people who have the time to criticize those who have the time to doodle, muse and conjecture.
okay the dude that complains about people doing this (anonymous 2 above) needs to get his medication, he must have missed a dosage or 3.
Wait - this guy is a logician and he let ten grand turn into pocket change??
the answer is 1.95,because 1week he gains 80%,and another he loses 60%,soo its 180 X 40% = 72%,that is his "gain" per 2 weeks
thus 365/7 = 52 gives u the number of weeks a year has,and 52/2 = 26 will let us know that this 72% actually repeats 26 times
so 10000 X (0.72^26) will gives u 1.95
i would say 140,000,
explainations are for girls :)
i would say 140,00
explanations are for boys :p :)
I gotta go with the $1.95.
Week one he makes 8,000 (giving him 18,000), the second week he lost $10,800 (18,000*-.6) and that gives him $7,200. Week 3 nets him an increase of $5,670 (totaling $12,870). Week 4 it drops down to $4,148, and so on, it's a case of diminishing returns it would seem to me. The only number that would fit is the $1.95
While the solution suggesting "a" is correct is elegant, it is nonetheless incorrect.
The problem is not worded in certainties; it is worded in probabilities. It says that there is a roughly 50% chance of each of the weekly possibilities (80% gain or 60% loss). The final statement, that you would know the remaining 26 weeks will be losses because the first 26 weeks were gains, is tantamount to saying that the next 26 flips of a coin would be tails because the first 26 were heads.
So, let's look at the right solution applied to a simpler example of the problem. Pretend this investment game only lasts two weeks. There are four possibilities (permutations) of outcomes: GG, GL, LG, and LL. That works out to three total combinations: 25% of the time, two gains will occur; 50% of the time, one gain and one loss will occur (as the solution presented indicates, multiplication is commutative, so GL and LG are equivalent); and 25% of the time, two losses will occur. Our expectation, therefore, is
$10,000 *
((1 / 4) * 1.8 * 1.8) +
(2 / 4) * 1.8 * 0.4 +
(1 / 4) * 0.4 * 0.4) = $12,100
In other words, 25% of the time, our $10,000 investment will become $32,400 (GG); 50% of the time, it will become $7,200 (GL or LG); and 25% of the time, it will become $1,600 (LL). The weighted average of these figures is $12,100. Because the difference between the best scenario ($32,400) and the typical scenario ($7,200) is much greater than the difference between the worst scenario ($1,600) and the typical scenario, the overall average is greater than the typical scenario.
Now, we can use this same logic to the problem presented, a 52-week period. The expectation in this case is $1,420,429.32. The correct answer, therefore, is D ($1.4 million).
My math must be off, my weighted average is only $9,850 using your numbers... 32400+2700+2700+1600=39400/4=9850.
that by 52 weeks is $512,200. (and the only number that would have gotten weighted more than the other numbers 2700 because it happens twice in your answer, the other numbers don't get weighted at all as they aren't more likely than the other).
well at least I see what I did wrong... I switched the 2's and the 7's around, oops
Eh, lol anonymous got me beat, my answer is wrong also after looking at his math and me not paying attention myself... lol sorry
I used a -.6 which would only be right if I subtracted the number I received from the first number, which I didn't do. Must...lay...off....sleep....depervation
To clarify the solution in my earlier post, the expression for the expectation in a 52-week period is
$10,000 *
(C(52, 0) / (2 ^ 52) * (1.8 ^ 52) * (0.4 ^ 0) +
C(52, 1) / (2 ^ 52) * (1.8 ^ 51) * (0.4 ^ 1) +
C(52, 2) / (2 ^ 52) * (1.8 ^ 50) * (0.4 ^ 2) +
...
C(52, 52) / (2 ^ 52) * (1.8 ^ 0) * (0.4 ^ 52)) = $1,420,429.32
where C(n, r) = n! / ((n - r)! * r!).
In response to Anon's comment:
"The problem is not worded in certainties; it is worded in probabilities. It says that there is a roughly 50% chance of each of the weekly possibilities (80% gain or 60% loss). The final statement, that you would know the remaining 26 weeks will be losses because the first 26 weeks were gains, is tantamount to saying that the next 26 flips of a coin would be tails because the first 26 were heads."
I agree with you that the problem is worded in such a way that you can't guarantee 26 losses in a row, but if it's truely random, then their are exactly 53 possible unique answers, half of which are below $1.95. That's why I limited my answer to:
"I'm assuming you meant 'exactly half the time', not 'about...' If the distribution of gains and losses are not equal, the answer becomes chaotic and essentially random after only 52 samples."
It was not the intent of the puzzle, but you might find this interesting.
Assuming you really do want a completely random distribution of gains and losses...
max loss is $10k (obviously)
--
max gain (52 gains in a row) is $188 quadrillion (although I doubt the bet could be paid off)
--
You are left with $1.95 11% of the time.
--
51% of the time you'll end up with between $0.09 and $39.54.
--
8.4% of the time you'll end up with a profit, sometimes a significant profit.
--
If 10M logisticians tried this, their combined average income would be about $740k. It's positive, btw, because you're risking at most $10k, but have virtually unlimited gains on the up-side. This assumes you will really be paid the enormous $$$ in the unlikely event that you get a huge number of gains with very few losses.
--
90% of the time you'll have less than $3604 after 52 weeks.
Oh, and btw, there is no combination of gains and losses that will net you exactly $1,420,429.32,
$14,000,
$140,000,
$1.4M, or
$131M
If 10 million logicians tried this investment scheme, their average income would not be $740,000. It would be the computed expectation: $1.4 million.
Expectation is a weighted average of all possible outcomes based upon their respective likelihoods of occurring. It should come as no surprise that no single outcome will yield the $1,420,429.32 figure I presented.
Think of the Deal or No Deal show. There are two cases left: one with $25,000 and one with $100,000. The banker offers you $60,000 to walk away. But your expectation in this situation is $62,500 because if you were to play the game hundreds of times in this very situation, your average per-game win would be the average of the two remaining amounts (it is equally likely for either one to be in the case you've chosen). So you should decline the offer because it is a bad deal. Again, even though you can't actually win exactly $62,500 in one iteration of this situation, it is nonetheless your expectation.
Note that one of the percentages quoted is off:
Only 6.3%, not 8.4%, of the time will you show a profit. The remaining 93.7% of the time will be a loss.
Thank you for your corrections. The error in my average income number was that I actually ran a simulation 10M times and many of the higher gains are so unlikely that they never occured and thus were not added to my results. Maybe if every person on the earth tried the scheme... :)
Heh. There are 2 ^ 52 = 4,503,599,627,370,496 permutations in this problem, so we'd probably need everyone on the planet to try the scheme several million times apiece to get a good distribution!
I have two solutions to this, one is you can't answer the question because if he gains/loses on his running total you would need to know the order in which he gains/loses. EX:
Week 1, $10,000 - lose 60% - now has $4,000
Week 2, 14,000 - Gain 80% - now has $25,200
-or-
Week 1, $10,000 - gain 80% - now has $18,000
Week 2, $18,000 - lose 60% - now has 7,200
Other solution is to solve it, 1 year = 52 weeks, half of the time (52/2=26) he gains 80%. so... 10,000*26*1.8 = 468,000(half his money)
10,000*26*.4 = 10,400 (the other half)
468,000+10,400 = 478,400
Bringing his total to $478,400
He could have kept the 10g a week and made 520,000. Lost 41,600. Sucker
laup - The order in which the gains/losses occur is actually irrelevant. There is a small mistake in your first calculation, in that if he lost 60% in week 1 he would only have $4000 to start week 2 - giving a result of $7200 the same as your second example.
The long and eloquent explanation from the other anonymous contributor is very entertaining but also wrong. This takes account of every possible outcome (and is correct for that scenario) but the question clearly states he makes a gain about half the time.
In your simple two week example both the LL and GG results do not fit this criteria as they are not winning about half the time.
The correct answer is in fact $195 which interestingly isn't provided as a possible choice, unless I am mistaken answer a) is $1.95 which is also wrong.
The expected return each week is 10%.
e.g. week 1.
Start with 10,000.
Win = 18,000.
Lose = 4,000.
Average = 11,000.
If we add 10% each week for 52 weeks we get the 1.4 million.
Week Total
1 £11,000.00
2 £12,100.00
3 £13,310.00
4 £14,641.00
5 £16,105.10
6 £17,715.61
7 £19,487.17
8 £21,435.89
9 £23,579.48
10 £25,937.42
11 £28,531.17
12 £31,384.28
13 £34,522.71
14 £37,974.98
15 £41,772.48
16 £45,949.73
17 £50,544.70
18 £55,599.17
19 £61,159.09
20 £67,275.00
21 £74,002.50
22 £81,402.75
23 £89,543.02
24 £98,497.33
25 £108,347.06
26 £119,181.77
27 £131,099.94
28 £144,209.94
29 £158,630.93
30 £174,494.02
31 £191,943.42
32 £211,137.77
33 £232,251.54
34 £255,476.70
35 £281,024.37
36 £309,126.81
37 £340,039.49
38 £374,043.43
39 £411,447.78
40 £452,592.56
41 £497,851.81
42 £547,636.99
43 £602,400.69
44 £662,640.76
45 £728,904.84
46 £801,795.32
47 £881,974.85
48 £970,172.34
49 £1,067,189.57
50 £1,173,908.53
51 £1,291,299.38
52 £1,420,429.32
I also verified the 1.4 million with a Monti Carlo simulation. it is correct.
He also loses money 93.6 % of the time.
Post a Comment
Links to this post:
Create a Link
<< Home