Probabilities

[Rat]

It's nearly four in the morning, and I'm absolutely hammered. I've just run a wine-tasting evening, and was hoping someone could help me with the probabilities. Basically, in a single-blinded test, contestants were given six whites and six reds, in numbered glasses, along with a list of all the wines that were presented.

I started to work out what number of correct guesses someone would need (I recognize that the sample size is small) to be significant. After a while, it occurred to me that the red and white tests could effectively be considered seperately, since they can see what they're drinking (I'm a bit slow), but that didn't help much.

Basically what I'm asking is how many someone would have to get right for it to be considered significant. But I'm very drunk. And I'm going to bed now.

Cheers.

 

[Unlike a Bull]

The statistical significance gets offset by the drunken insignificance. So, for example, if one were to have to drink all six glasses of wine to reach statistical significance, one would stop caring by the time the sixth glass was reached. This is known as the "shut up, I don't care" law.










I'm sure someone will be by later with actual math and numbers and such.

 

[DevilsAdvocate]

I assume they were told that they would be given six wines and then the glasses were mixed up and they had to guess which one was which. Correct? If so, I can say unquestionably that if anyone got exactly five right, they would qualify for the MDC!<sup>*



*</sup>Assuming they didn't guess the same wine for more than one glass.

 

[DevilsAdvocate]

6 wines. 6 glasses. Non-repetitive.
6_P_6
Number of permutations is 6!.
6*5*4*3*2*1=720 permutations.

Now we use the inclusion-exclusion principal to count the derangements.

Chances for getting EXACTLY:

0​

|

265​

|

36.81%

1​

|

264​

|

36.67%

2​

|

135​

|

18.75%

3​

|

40​

|

5.56%

4​

|

15​

|

2.08%

5​

|

0​

|

0.00%

6​

|

1​

|

0.14%

Total​

|

720​

|

100.00%​

Chances for getting AT LEAST:

0​

|

720​

|

100.00%

1​

|

455​

|

63.19%

2​

|

191​

|

26.53%

3​

|

56​

|

7.78%

4​

|

16​

|

2.22%

5​

|

1​

|

0.14%

6​

|

1​

|

0.14%​

So the odds for getting at least 4 right is only 1 in 45.

Then we can combine the two tests together. The number of permutations is 720*720 = 518,400.

Chances for getting EXACTLY:

0​

|

70225​

|

13.5465%

1​

|

139920​

|

26.9907%

2​

|

141246​

|

27.2465%

3​

|

92480​

|

17.8395%

4​

|

47295​

|

9.1233%

5​

|

18720​

|

3.6111%

6​

|

6180​

|

1.1921%

7​

|

1728​

|

0.3333%

8​

|

495​

|

0.0955%

9​

|

80​

|

0.0154%

10​

|

30​

|

0.0058%

11​

|

0​

|

0.0000%

12​

|

1​

|

0.0002%​

Chances for getting AT LEAST:

0​

|

518400​

|

100.0000%​

|

1 in 1.0

1​

|

448175​

|

86.4535%​

|

1 in 1.2

2​

|

308255​

|

59.4628%​

|

1 in 1.7

3​

|

167009​

|

32.2162%​

|

1 in 3.1

4​

|

74529​

|

14.3767%​

|

1 in 7.0

5​

|

27234​

|

5.2535%​

|

1 in 19.0

6​

|

8514​

|

1.6424%​

|

1 in 60.9

7​

|

2334​

|

0.4502%​

|

1 in 222.1

8​

|

606​

|

0.1169%​

|

1 in 855.4

9​

|

111​

|

0.0214%​

|

1 in 4670.3

10​

|

31​

|

0.0060%​

|

1 in 16722.6

11​

|

1​

|

0.0002%​

|

1 in 518400.0

12​

|

1​

|

0.0002%​

|

1 in 518400.0​

So at least 8 correct would be about 1 in 855. Geting at least 9 correct would be about 1 in 4673.

 

[Squeegee Beckenheim]

So at least 8 correct would be about 1 in 855. Geting at least 9 correct would be about 1 in 4673.

Probability was never my strong suit, but how can you get 8 or 9 right out of a possible 6?

 

[Tubbythin]

I assume they were told that they would be given six wines and then the glasses were mixed up and they had to guess which one was which. Correct? If so, I can say unquestionably that if anyone got exactly five right, they would qualify for the MDC!^* .

I may have missed the point here, but when did being a wine conniosseur qualify someone for the MDC?

 

[Professor Yaffle]

If you get 5 right and don't repeat a guess, the sixth will be correct too.

 

[Professor Yaffle]

Probability was never my strong suit, but how can you get 8 or 9 right out of a possible 6?

6 reds and 6 whites... So up to 12 correct answers.

 

[Tubbythin]

If you get 5 right and don't repeat a guess, the sixth will be correct too.

Was that a response to me?

 

[Professor Yaffle]

Was that a response to me?

Yes.

 

[Aepervius]

And so DevilAdvocate win the thread .

@Rat out of curiosity will you give the results of the blind test ?

 

[Tubbythin]

Yes.

Oops. I did miss something. Specifically I missed the "exactly" in the quotation. Instead my head seems to have read "at least". I feel a bit stupid now.

 

[Squeegee Beckenheim]

6 reds and 6 whites... So up to 12 correct answers.

Ah, I misread the OP - I thought it was 3 reds and 3 whites making a total of 6.

Thanks.

 

[DevilsAdvocate]

Oops. I did miss something. Specifically I missed the "exactly" in the quotation. Instead my head seems to have read "at least". I feel a bit stupid now.

While waiting to see if anyone else posted a solution, I was having some fun with the statistics (hence the big grinny face). It can be intuitive to think that 6 is too difficult, but that 3 or 4 is too easy, so maybe getting 5 right would be significant. Of course, if you think about it, you can’t get exactly 5 right. If you get 4 right, then the other 2 are either switched around wrong (meaning you got 4 right) or are correct (meaning you got 6 right). You can’t get exactly 5 right.

It was a bit of a joke, but it is an important concept to this type of test. The chances of getting 6 right out of 6 is only 1 in 720. But what if one of the wines is easily distinguishable? If the subject can easily place that wine, then guess the rest drops to 1 in 120. And if the subject can easily identify 2 of the wines, the chances of correctly ordering the remaining 4 is just 1 in 24. The chances of success increases much more rapidly than we might initially assume.

 

[Rat]

The statistical significance gets offset by the drunken insignificance. So, for example, if one were to have to drink all six glasses of wine to reach statistical significance, one would stop caring by the time the sixth glass was reached. This is known as the "shut up, I don't care" law.

Fortunately, while I said 'glasses', they didn't actually get whole glasses of each. They got a small amount of each, and then were allowed to try more of the ones they were unsure about, or just that they particularly liked, with dinner. So while I think everyone was drunk by the end of the night, they had tasted each wine before being so.

 

[Rat]

Chances for getting AT LEAST:

0​

|

518400​

|

100.0000%​

|

1 in 1.0

1​

|

448175​

|

86.4535%​

|

1 in 1.2

2​

|

308255​

|

59.4628%​

|

1 in 1.7

3​

|

167009​

|

32.2162%​

|

1 in 3.1

4​

|

74529​

|

14.3767%​

|

1 in 7.0

5​

|

27234​

|

5.2535%​

|

1 in 19.0

6​

|

8514​

|

1.6424%​

|

1 in 60.9

7​

|

2334​

|

0.4502%​

|

1 in 222.1

8​

|

606​

|

0.1169%​

|

1 in 855.4

9​

|

111​

|

0.0214%​

|

1 in 4670.3

10​

|

31​

|

0.0060%​

|

1 in 16722.6

11​

|

1​

|

0.0002%​

|

1 in 518400.0

12​

|

1​

|

0.0002%​

|

1 in 518400.0​

So at least 8 correct would be about 1 in 855. Geting at least 9 correct would be about 1 in 4673.

That is exactly what I was looking for, but was too stupid (and drunk) to work out. Cheers.

The winner managed to get 4 right, consisting of two reds and two whites. So since there were 4.5 people playing, and the odds of getting 4 right are 1 in 7, nobody did spectacularly well. I say 4.5 people playing, since I took part but saw a couple of the bottles being wrapped, so I was not properly blinded. I scored 4, but two of those I knew.

The most expensive (a Châteauneuf-du-Pape) was not highly rated, and nobody identified it. Those who tasted it after the bottles were revealed said it now tasted much better, including someone who often drinks it and normally likes it. The cheapest was Sainsbury's Basics Spanish Red, which is £2.74, about the cheapest you can get a wine in this country. Nobody identified it and everybody liked it.

The only one that nearly everyone got was the Riesling, because of course it was very sweet. I made a mistake getting that, because I wanted a German white, but couldn't find a dry one, and all the other whites in the test were dry or medium-dry.

On the whole, people did badly, but slightly better than the whisky/whiskey tasting test I did last year, where almost everyone did worse than chance would predict.

 

[sol invictus]

The most expensive (a Châteauneuf-du-Pape) was not highly rated, and nobody identified it. Those who tasted it after the bottles were revealed said it now tasted much better, including someone who often drinks it and normally likes it. The cheapest was Sainsbury's Basics Spanish Red, which is £2.74, about the cheapest you can get a wine in this country. Nobody identified it and everybody liked it.

I read something recently that claimed that people systematically prefer cheaper wines in blind tests - even when the subjects are wine industry professionals.

On the whole, people did badly, but slightly better than the whisky/whiskey tasting test I did last year, where almost everyone did worse than chance would predict.

I'm surprised by that - anyone can tell Laphroig from Glenlivet. But I guess it depends on which whiskeys you used.

 

[Rat]

I'm surprised by that - anyone can tell Laphroig from Glenlivet. But I guess it depends on which whiskeys you used.

And indeed most people got Laphroaig, but otherwise did very badly. It was a mix of single malts and blends, from the cheapest to the fairly expensive and varying regions, mostly around Scotland but with one Irish and one Canadian. The winner got, I think, two right. Among people most of whom would consider themselves whisky connoisseurs, they weren't even particularly impressive at judging whether they were drinking a malt or a blend. And if you think you could do better, I'd urge you to try a proper DBT.

 

[sol invictus]

And indeed most people got Laphroaig, but otherwise did very badly. It was a mix of single malts and blends, from the cheapest to the fairly expensive and varying regions, mostly around Scotland but with one Irish and one Canadian. The winner got, I think, two right. Among people most of whom would consider themselves whisky connoisseurs, they weren't even particularly impressive at judging whether they were drinking a malt or a blend. And if you think you could do better, I'd urge you to try a proper DBT.

I once had a friend select 4 out of about 7-8 single malts in my liquor cabinet. I got all 4 not knowing which 4 she had chosen. But it helped tremendously that these were mine, so I was very familiar with them.

Another time we tried a blind tasting of unflavored vodkas. It was a total failure - I think they all taste exactly the same (but I thought that before the test too).

 

[Ray Brady]

I was toying with this problem, and came up with numbers different from what DA offers. Can anyone explain where I'm going wrong?

I started with the assumption that the odds of getting at least one guess out of six right would be equal to 1 minus the odds of getting none right. To determine this, I figured:

The odds of getting the first guess wrong are 5/6, or 83%.
Since there are now only five options to choose from, the odds of getting the second guess wrong are now 4/5, or 80%. Continuing on in this way, I would assume that the odds of getting the third, fourth, fifth and sixth guesses wrong are 3/4, 2/3, 1/2 and 1, respectively.

Multiplying 83% * 80% * 75% * 67% * 50% * 100% gives me 16.7%. Thus, the odds of getting at least one right should be 83.3%, though DA says it's actually 63.2%.

Have I made an error here somewhere?