Solve this math problem and win a prize (expired)

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Damn Getch, you're right.

Add me and tomorrow morning (when I am back online) and I will give you something else (already gave away the other stuff).

He's not right.

The answer is 50%....even though Azaroth didnt word it clearly.

Paraphrasing Azaroth - the chance of opening the correct portal is 100%. Would have been different if you specifically asked, say, what are the odds opening it on the 2nd machine. But you didnt. You asked what are the odds of opening the machine. It's 100% since you have 3 machines.

So the odds of getting the organ to drop is 50%.

Dont let Cyberbeni's diagrams and number crunching confuse you.


This is only true if you open all 3 portals in the same game no matter what. But that is hardly the optimal strategy. =)
1/2*1/2*1/2 = 1/8
87.5% is way off actually. It doesn't take into account the chances of getting the right portal. It's the chance of getting the organ if all of your portals roll the right pair of bosses.

I did my analysis this way:

Chance of winning on first try = chance of getting right portal * chance of getting the drop
= 1/3 * 1/2 = 1/6. Since we'll be dealing with lots of 1/3 and 1/2, let's use a common denominator of 216 (6^3) so this is 36/216.

Chance of winning on second try is complicated. You have a 2/3 chance of having gotten the wrong portal on the first try, followed by a 50% chance of getting the right portal on the second try (you've already eliminated one of the wrong choices so your chance of getting the right portal is better after a "wrong portal"), followed by a 50% chance of getting the drop which is 2/3*1/2*1/2 = 2/12 = 1/6 = 36/216.

But, you also had a 1/6 chance of getting the right portal on the first try, but not getting the drop, then quitting the game and restarting, then having a 1/3 chance of getting the right portal on the second try and a 1/2 chance of getting the drop for 1/6*1/3*1/2 = 1/36 = 6/216.

It gets complicated on missing it twice and winning on the third try. You have the following list of possibilities. I'll rewrite the "win in 1" and "win in 2" cases here for consistency:

Y = got right portal and got organ
n = got right portal and did not get organ
a = got wrong portal #1
b = got wrong portal #2

win in 1:
Y = (1/3 * 1/2) = 1/6 = 36/216

win in 2:
a Y = 1/3 * (1/2 * 1/2) = 1/12 = 18/216
b Y = 18/216
n Y = (1/3 * 1/2) * (1/3 * 1/2) = 1/36 = 6/216

win in 3:
a b Y = 1/3 * 1/2 * (1 * 1/2) = 1/12 = 18/216
b a Y = 1/3 * 1/2 * (1 * 1/2) = 18/216
a n Y = 1/3 * (1/2 * 1/2) * (1/3 * 1/2) = 1/72 = 3/216
b n Y = 3/216
n a Y = (1/3 * 1/2) * 1/3 * (1/2 * 1/2) = 1/72 = 3/216
n b Y = 3/216
n n Y = 1/6 * 1/6 * 1/6 = 1/216

Sum = (36 + 18+18+6 + 18+18+3+3+3+3+1)/216
= (36 + 42 + 49)/216
= 127/216
= 0.587962962962963

To verify empirically, the chances of winning are .5879 according to a test program I just wrote.
87.5% is way off actually. It doesn't take into account the chances of getting the right portal. It's the chance of getting the organ if all of your portals roll the right pair of bosses.

I did my analysis this way:

Chance of winning on first try = chance of getting right portal * chance of getting the drop
= 1/3 * 1/2 = 1/6. Since we'll be dealing with lots of 1/3 and 1/2, let's use a common denominator of 216 (6^3) so this is 36/216.

Chance of winning on second try is complicated. You have a 2/3 chance of having gotten the wrong portal on the first try, followed by a 50% chance of getting the right portal on the second try (you've already eliminated one of the wrong choices so your chance of getting the right portal is better after a "wrong portal"), followed by a 50% chance of getting the drop which is 2/3*1/2*1/2 = 2/12 = 1/6 = 36/216.

But, you also had a 1/6 chance of getting the right portal on the first try, but not getting the drop, then quitting the game and restarting, then having a 1/3 chance of getting the right portal on the second try and a 1/2 chance of getting the drop for 1/6*1/3*1/2 = 1/36 = 6/216.

It gets complicated on missing it twice and winning on the third try. You have the following list of possibilities. I'll rewrite the "win in 1" and "win in 2" cases here for consistency:

Y = got right portal and got organ
n = got right portal and did not get organ
a = got wrong portal #1
b = got wrong portal #2

win in 1:
Y = (1/3 * 1/2) = 1/6 = 36/216

win in 2:
a Y = 1/3 * (1/2 * 1/2) = 1/12 = 18/216
b Y = 18/216
n Y = (1/3 * 1/2) * (1/3 * 1/2) = 1/36 = 6/216

win in 3:
a b Y = 1/3 * 1/2 * (1 * 1/2) = 1/12 = 18/216
b a Y = 1/3 * 1/2 * (1 * 1/2) = 18/216
a n Y = 1/3 * (1/2 * 1/2) * (1/3 * 1/2) = 1/72 = 3/216
b n Y = 3/216
n a Y = (1/3 * 1/2) * 1/3 * (1/2 * 1/2) = 1/72 = 3/216
n b Y = 3/216
n n Y = 1/6 * 1/6 * 1/6 = 1/216

Sum = (36 + 18+18+6 + 18+18+3+3+3+3+1)/216
= (36 + 42 + 49)/216
= 127/216
= 0.587962962962963

To verify empirically, the chances of winning are .5879 according to a test program I just wrote.


Correct.

Its a binomial distribution.
Find the probability of winning (p)
Find q which is 1 - p
Then use nCr on your calculator to find the probability using the binomial formula
(NCR)(p^r)(q^(n-r))

the quote above is doing it the long way using a tree diagram but same thing.
ALso the original question was not clear. Re-write the question so people know exactly what you are after. (i.e. what probability you are after).
this thread is still going?

The correct answer to this problem is:
ask someone for an MP10 run
who thinks festavus should have won?
07/23/2013 03:46 AMPosted by Mowze
who thinks festavus should have won?

No, the first correct response is on page two by Getch:

Brute force, of all possible outcomes. "Good" results in bold:
...[snip]...
1/6 + 1/36 * 1/216 + 1/36 + 1/6 + 1/36 + 1/6 = 127/216 = 58.8%
The simple answer is there is no answer. It all comes down to the simple coin toss, 50/50 chance at getting heads or tails but 10 coin flips in a row could all be tails meaning a 50/50 chance ends in 100% 1 sided result. So basically it comes down to luck, no mathematical work will result in a correct answer no matter how smart you think you are : )
removed
Game Portal Loot Chance
1______1______0.166666667
2______1______0.027777778
2______2______0.027777778
3______1______0.00462963
1______2______0.166666667
2______1______0.027777778
1______3______0.166666667
0.587962963
as many have pointed.127 / 216!

However instead of doing it on paper or manually. We can do it with a program. A program can produce the result for a variable number of machines. Also we can determine the average organ yield where a user will use all machines in the search to maximize 1 organ type.

source:
http://pastebin.com/raw.php?i=fmwmBebd

For 1 machines, Chance To Get >=1 Organ
1 / 6 ~= 0.16666666666666666
For 1 machines, Average Organs from spending all machines
1 / 6 ~= 0.16666666666666666
For 2 machines, Chance To Get >=1 Organ
13 / 36 ~= 0.3611111111111111
For 2 machines, Average Organs from spending all machines
7 / 18 ~= 0.3888888888888889
For 3 machines, Chance To Get >=1 Organ
127 / 216 ~= 0.5879629629629629
For 3 machines, Average Organs from spending all machines
37 / 54 ~= 0.6851851851851852
For 4 machines, Chance To Get >=1 Organ
889 / 1296 ~= 0.6859567901234568
For 4 machines, Average Organs from spending all machines
74 / 81 ~= 0.9135802469135802
For 5 machines, Chance To Get >=1 Organ
6007 / 7776 ~= 0.7725051440329218
For 5 machines, Average Organs from spending all machines
565 / 486 ~= 1.162551440329218
For 6 machines, Chance To Get >=1 Organ
39241 / 46656 ~= 0.8410708161865569
For 6 machines, Average Organs from spending all machines
2071 / 1458 ~= 1.4204389574759946
For 7 machines, Chance To Get >=1 Organ
247255 / 279936 ~= 0.8832554583904892
For 7 machines, Average Organs from spending all machines
7285 / 4374 ~= 1.6655235482395976
For 8 machines, Chance To Get >=1 Organ
1538761 / 1679616 ~= 0.916138569768328
For 8 machines, Average Organs from spending all machines
12572 / 6561 ~= 1.91617131534827
For 9 machines, Chance To Get >=1 Organ
9473815 / 10077696 ~= 0.9400774740575624
For 9 machines, Average Organs from spending all machines
85321 / 39366 ~= 2.167377940354621
For 10 machines, Chance To Get >=1 Organ
57840649 / 60466176 ~= 0.95657858370273
For 10 machines, Average Organs from spending all machines
285367 / 118098 ~= 2.416357601314163
For 15 machines, Chance To Get >=1 Organ
466248612439 / 470184984576 ~= 0.9916280352071436
For 15 machines, Average Organs from spending all machines
105225877 / 28697814 ~= 3.6666861455022324
For 20 machines, Chance To Get >=1 Organ
3650240655886153 / 3656158440062976 ~= 0.9983814202054326
For 20 machines, Average Organs from spending all machines
17143360514 / 3486784401 ~= 4.916667778220911
For 25 machines, Chance To Get >=1 Organ
28421393604598182679 / 28430288029929701376 ~= 0.9996871496580634
For 25 machines, Average Organs from spending all machines
10449892698457 / 1694577218886 ~= 6.166666577358255
For 30 machines, Chance To Get >=1 Organ
221060551987010497582729 / 221073919720733357899776 ~= 0.9999395327420811
For 30 machines, Average Organs from spending all machines
3054051791156359 / 411782264189298 ~= 7.416666662827418
For 35 machines, Chance To Get >=1 Organ
1719050708823285319818255319 / 1719070799748422591028658176 ~= 0.9999883129158263
For 35 machines, Average Organs from spending all machines
867213448422599077 / 100063090197999414 ~= 8.66666666706579
For 40 machines, Chance To Get >=1 Organ
13367464343333472198458637923785 / 13367494538843734067838845976576 ~= 0.9999977411241745
For 40 machines, Average Organs from spending all machines
120563515802466689996 / 12157665459056928801 ~= 9.916666666679182
For 45 machines, Chance To Get >=1 Organ
103945592151930507026628494195881111 / 103945637534048876111514866313854976 ~= 0.9999995634052622
For 45 machines, Average Organs from spending all machines
65979650446291639208989 / 5908625413101667397286 ~= 11.166666666664922
For 50 machines, Chance To Get >=1 Organ
808281209258046236569910278620995970313 / 808281277464764060643139600456536293376 ~= 0.999999915615121
For 50 machines, Average Organs from spending all machines
17827800027680952638267623 / 1435795975383705177540498 ~= 12.41666666666663
For 55 machines, Chance To Get >=1 Organ
6285195111055225026007329091629320359394647 / 6285195213566005335561053533150026217291776 ~= 0.9999999836901201
For 55 machines, Average Organs from spending all machines
4768278434249287507399132213 / 348898422018240358142341014 ~= 13.666666666666675
For 60 machines, Chance To Get >=1 Organ
48873677826621441059611310490774565486657545289 / 48873677980689257489322752273774603865660850176 ~= 0.999999996847632
For 60 machines, Average Organs from spending all machines
632334777605308372953355674494 / 42391158275216203514294433201 ~= 14.916666666666666
For 65 machines, Chance To Get >=1 Organ
380041719746284585036848352135758529770884477187607 / 380041719977839666236973721680871319659378770968576 ~= 0.9999999993907114
For 65 machines, Average Organs from spending all machines
333067330568373710361443565523009 / 20602102921755074907947094535686 ~= 16.166666666666664
For 70 machines, Chance To Get >=1 Organ
2955204414199667253988670995034753366738871248409031561 / 2955204414547681244658707659790455381671329323051646976 ~= 0.999999999882237
For 70 machines, Average Organs from spending all machines
87193250090597915778581373076618135 / 5006311009986483202631143972171698 ~= 17.416666666666664
For 75 machines, Chance To Get >=1 Organ
22979669526999724314174900030133295952727806488083999792855 / 22979669527522769358466110762530581047876256816049606885376 ~= 0.9999999999772387
For 75 machines, Average Organs from spending all machines
22708626741298687807294063867776251269 / 1216533575426715418239367985237722614 ~= 18.666666666666664
For 80 machines, Chance To Get >=1 Organ
178689910245230947864195115025159271684561039058833158353113801 / 178689910246017054531432477289437798228285773001601743140683776 ~= 0.9999999999956007
For 80 machines, Average Organs from spending all machines
2943859185835722972711901601176431312440 / 147808829414345923316083210206383297601 ~= 19.916666666666664
For 85 machines, Chance To Get >=1 Organ
1389492742071847142855438058262361913913392752242838344357906462615 / 1389492742073028616036418943402668319023150170860455154661957042176 ~= 0.9999999999991497
For 85 machines, Average Organs from spending all machines
1520509428185376513152509655062636473932677 / 71835091095372118731616440160302282634086 ~= 21.166666666666664
For 90 machines, Chance To Get >=1 Organ
10804695562358094831950822343985392873842673139349587960058408190231049 / 10804695562359870518299193703899148848724015728610899282651377959960576 ~= 0.9999999999998358
For 90 machines, Average Organs from spending all machines
391303699969265773760797773574888407606785623 / 17455927136175424851782794958953454680082898 ~= 22.416666666666668
For 95 machines, Chance To Get >=1 Organ
84017312692907684395676524715943347762122344351729656658870896059353972823 / 84017312692910353150294530241519781447677946305678352821897115016653438976 ~= 0.9999999999999681
For 95 machines, Average Organs from spending all machines
100389036960144868322602862882902248395663005909 / 4241790294090628238983219175025689487260144214 ~= 23.666666666666664
For 100 machines, Chance To Get >=1 Organ
653318623500066895111744835624186667667445448787330805035663512014523572535625 / 653318623500070906096690267158057820537143710472954871543071966369497141477376 ~= 0.9999999999999939
For 100 machines, Average Organs from spending all machines
12841489891572615664991823126350166442442489160218 / 515377520732011331036461129765621272702107522001 ~= 24.916666666666668
For 1000 machines, Chance To Get >=1 Organ
14166102623834861723796252524915224416640471830910191322323547432140618947596486436347661333869287260068907949302029484915942402681211620694597762903768493031793437496360887770405170017360367775685180887600222888037722079533459491004747944372063257507753971205889900009748316822012021763199215894753285262008784429860791005107434776049420802546396387214484156773576780563603487216193121989408145776439986674339488257614492579842146276269121340448272733184321294740548634134183983966542587002322710193226457256012433516522674636684321859849905433686605500321425343345293648345146344534788883625829081980922149300258663737339530754547325785844790709651498911546241790096022163928512788463291211005181604538995905550137997898318998598275884992295341662153949271529602704581957807561 / 14166102623834861723796252524915224416640471830910191322323547432140618947596486436347661333869287260068907949302029484915942402681211620694598046617844295512220793103312980549591537160959053027940624117598003417503015722697428176155600362263128567590299511776686592862074376328232990325101248680123776914576482815095784568122986221890411837737570098864613342090972756469661488216176894465388028416768338495326989675118087222767384596111351304957869025273802978281783731929966468210579229830069556698928937342508988340792335737744719376598506908977135291983117722648269177947154657697517074993441515526839887073400191797445153760221695723268255006134044062503100710134200414607696976757837002911389023284338696251543694980946202137938610119300450795091488653253649628649410789376 ~= 1.0
For 1000 machines, Average Organs from spending all machines
330407532301911592002872943666390079460858389685307696235591348692688246953563999476899513603164464819147118609410072873791409399309979676390346902754050255700736087947480340902513858992722129915158067496054022125445476208254471684978886412751442208008396954913448456451387654592148260052196293963508064448695715170758699493425828257141535393054628262870186955330897089997673076772504054490728597779409252876907260634556731276929839164826137380779583931731826448386377563191357036 / 1322070819480806636890455259752144365965422032752148167664920368226828597346704899540778313850608061963909777696872582355950954582100618911865342725257953674027620225198320803878014774228964841274390400117588618041128947815623094438061566173054086674490506178125480344405547054397038895817465368254916136220830268563778582290228416398307887896918556404084898937609373242171846359938695516765018940588109060426089671438864102814350385648747165832010614366132173102768902855220001 ~= 249.91666666666666
07/23/2013 04:15 AMPosted by Nixsta
The simple answer is there is no answer. It all comes down to the simple coin toss, 50/50 chance at getting heads or tails but 10 coin flips in a row could all be tails meaning a 50/50 chance ends in 100% 1 sided result. So basically it comes down to luck, no mathematical work will result in a correct answer no matter how smart you think you are : )


The simple answer is that there is an answer; and you, my friend, definitely aren't very smart, no matter what you think...
The real answer is 200% because he is multiboxing 4 characters.
And the final answer is that this has already been solved and is over with. LOL

Sheesh.
1/3*1/2+1/2*1/2+1/2=1/6+1/4+1/2=11/12
Getch was the first one to get the answer right, so I still owe him a prize. I gave the first prize to someone else, mainly because I was ready for the thread to die.

I knew from the get go that the odds were better than 50%, because there are more variables thrown into the equation than just 3 portals, 1 game. I wasn't sure of the exact figure, but I knew it was better than 50%. I was swayed a little bit by the number of people insisting on 50%, but I am glad that the problem was resolved and my initial hunch was proven correct (hence the reason for the question).

In any case, thanks to everyone who participated. I owe Getch a prize and if he doesn't add me by the end of today I will give the gold to the first person who confirmed Getch's answer. My ignorance of algebraic statistics was the cause for my asking the work to be shown, but after carefully looking at the solution, it's clear that .58 is the correct answer.

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