Escaped cow puzzle 1

[md65536]

A farmer (on an island of course) has 100 cows in a barn.

Each is brown or blue, and each was purchased from a supplier that randomly sells a brown or a blue cow with equal likelihood.

 

A cow escapes from the barn and it is brown.

What is the expected proportion of brown cows to total cows?

Edited June 22, 2012 by md65536

[J.C.MacSwell]

Since you could not have a half brow half blue cow, 50 or 51% is the most likely

[EquisDeXD]

A farmer (on an island of course) has 100 cows in a barn.

Each is brown or blue, and each was purchased from a supplier that randomly sells a brown or a blue cow with equal likelihood.

 

A cow escapes from the barn and it is brown.

What is the expected proportion of brown cows to total cows?

 

 

100% brown cows because blue cows don't exist

 

[GlassPilot]

It's still 50:50. Taking a brown cow out of the mix doesn't change the likelihood or chance of any cow being brown or blue.

Edited October 17, 2012 by GlassPilot

[md65536]

 

It's still 50:50. Taking a brown cow out of the mix doesn't change the likelihood or chance of any cow being brown or blue.

 

No, that's not correct. The likelihood of the one brown cow being brown is 100% certain.

 

Since you could not have a half brow half blue cow, 50 or 51% is the most likely

I'll repeat my reply from the other thread because it's more appropriate here: Unless I worded it wrong, the expected proportion will be the mean of all equally probable possible proportions. The mean can be fractional. You've given the mode.

[imatfaal]

I would look at it this way - although I do tend to forsake the obvious

 

1. There are 2^n distributions of white and brown cows (I have never seen a blue cow, i never hope to see one...)

2. 2^n -1 of these allow the removal of one brown cow (there is one combo that is only white)

3. Of the combinations which allow, after the removal of one brown cow there are a total number of brown cows in all possible combos equal to

[math] \sum^{n-1}_{k} \frac{n!}{k!(n-k)!}(n-k-1)[/math]

4. And the total number of cows in all combos of

[math](2^n-1)(n-1)[/math]

5. divide one by tother

 

numerically this is two 30 odd digit numbers - but it reduces to 49/99

 

not sure if I have taken a wrong step but the answer seems awfully naive

[md65536]

I would look at it this way - although I do tend to forsake the obvious

 

1. There are 2^n distributions of white and brown cows (I have never seen a blue cow, i never hope to see one...)

2. 2^n -1 of these allow the removal of one brown cow (there is one combo that is only white)

3. Of the combinations which allow, after the removal of one brown cow there are a total number of brown cows in all possible combos equal to

[math] \sum^{n-1}_{k} \frac{n!}{k!(n-k)!}(n-k-1)[/math]

4. And the total number of cows in all combos of

[math](2^n-1)(n-1)[/math]

5. divide one by tother

 

numerically this is two 30 odd digit numbers - but it reduces to 49/99

 

No, that's not correct. Note that 49/99 is less than half. Why would finding out that one of the cows is brown make you think that less than half of the cows are brown?

 

I'm not sure where the mistake is... perhaps step 4? Why is it multiplied by (n-1) instead of n?

 

BTW the answer is much much easier to figure out!

 

 

[imatfaal]

No, that's not correct. Note that 49/99 is less than half. Why would finding out that one of the cows is brown make you think that less than half of the cows are brown?

 

 

I'm not sure where the mistake is... perhaps step 4? Why is it multiplied by (n-1) instead of n?

 

BTW the answer is much much easier to figure out!

 

 

Do you mean the proportion of the 100 cows or of the cows that are left? IF it is the cows that are left - which the other cow being lost escaping implies - then of course there will be a lower than half expectation of the proportion of cows that are brown. do it with three cows at the start - out of the three possible scenarios in which you remove a cow two will leave you with a white cow and one with a brown cow

[md65536]

Do you mean the proportion of the 100 cows or of the cows that are left? IF it is the cows that are left - which the other cow being lost escaping implies - then of course there will be a lower than half expectation of the proportion of cows that are brown. do it with three cows at the start - out of the three possible scenarios in which you remove a cow two will leave you with a white cow and one with a brown cow

I meant the proportion of the 100 cows (the farmer still has 100 cows), but I wasn't very clear. But still, the answer you gave isn't correct for the proportion of cows in the barn either.

 

It it essentially a random cow that escapes, which happens to be brown. It is not selected to be brown (see Escaped cow puzzle 2 for that version!). The color of each cow is independent, meaning that the color of the n'th cow bought doesn't depend on what the previously bought cow colors are.

 

In your counterexample, there are 4 possible scenarios that are equally probable. You've counted two cases as a single case, which make 3 possible scenarios which are not all equally probable.

 

 

[imatfaal]

I meant the proportion of the 100 cows (the farmer still has 100 cows), but I wasn't very clear. But still, the answer you gave isn't correct for the proportion of cows in the barn either.

 

It it essentially a random cow that escapes, which happens to be brown. It is not selected to be brown (see Escaped cow puzzle 2 for that version!). The color of each cow is independent, meaning that the color of the n'th cow bought doesn't depend on what the previously bought cow colors are.

 

In your counterexample, there are 4 possible scenarios that are equally probable. You've counted two cases as a single case, which make 3 possible scenarios which are not all equally probable.

 

 

 

Not true - the escaped cow gives you information. I mistyped my counter - never a good start! I meant to say take the example with two cows at start as extreme example

BB ->B

BW ->W

WB ->W

WW XX does not happen

 

You have three combinations from your original four that still fit (if a brown cow exists to escape then your initial selection could not have been WW). The remaining three scenarios have a total of three cows of which 2 are white - my expectation is not even for brown and white cows

 

For a three Cow scenario (n=3) you can see that the expectation ratio changes

BBB -> BB

 

BBW -> BW

BWB -> BW

WBB -> BW

 

WWB -> WW

WBW -> WW

BWW -> WW

 

WWW XX does not happen.

 

You now have a total of 7 (2^n-1) scenarios that feasibly follow from initial conditions and 14 cows { 2^n-1)(n-1) . 5 are brown and 9 are white - expectations will always remain uneven - because you have lost a cow.

 

For the expectation ratio of the entire 100 cows you are still incorrect. The fact that at least one cow is brown is added information and rules out the scenario that all the cows were white. Looking at the LHS column above - you have 7 scenarios, the total number of cows in all scenarios is 21, 9 of which are white and 12 of which are brown.

 

Change your expectation of the total number into a game of chance and black and white balls - three balls in a bag, each could be black or white with equal probability; you are shown that one is black (it is replaced) and then have to put a fiver on the colour of the next ball drawn. Would you be willing to put your fiver on white? Cos I would go with black every time.

 

And if the ball is not replaced I would go with white every time. The fact that a black ball exists changes the scenarios that you have to consider by removing one.

[md65536]

Not true - the escaped cow gives you information. I mistyped my counter - never a good start! I meant to say take the example with two cows at start as extreme example

BB ->B

BW ->W

WB ->W

WW XX does not happen

 

If you do it this way then you also have to take into consideration the probability that a random selection of one will be B. That is 1.0, 0.5, 0.5 (and 0.0) respectively.

 

Another way to achieve the same result is to take the first of the pair as the escaped cow. Like you allow, case WW doesn't happen. If you do it this way, neither does WB.

 

I worded the puzzle to be sure not to imply that a brown cow was specifically selected for its color, to escape. In your example, what is the reasoning for choosing the first cow in the BW case, but the second in the WB case?

 

 

 

Change your expectation of the total number into a game of chance and black and white balls - three balls in a bag, each could be black or white with equal probability; you are shown that one is black (it is replaced) and then have to put a fiver on the colour of the next ball drawn. Would you be willing to put your fiver on white? Cos I would go with black every time.

Agreed; this is what I meant when I said that the knowledge of a brown cow should not make one think that there will be fewer than 50% brown cows.

 

 

Try your examples with coin tosses. Suppose you toss a coin and it is heads. What will the next toss more likely be?

[imatfaal]

MD6

 

Gotta head for the hills - but very interesting response. Will draft and reply tomorrow. Thanks

[imatfaal]

I am now almost entirely convinced you are correct - but I want to run sum simulations. I have no real grasp of probabilities and I have run the gamut from MD65336 is talking nonsense through to I am talking nonsense - so I will do a bit more testing.

[md65536]

I am now almost entirely convinced you are correct - but I want to run sum simulations. I have no real grasp of probabilities and I have run the gamut from MD65336 is talking nonsense through to I am talking nonsense - so I will do a bit more testing.

From a stats perspective the answer is extremely simple...

 

 

 

Half of the 99 unknown cows are expected to be brown, plus the one known brown cow makes (49.5 + 1)/100 = 50.5% brown cows expected.

 

To see that this is true, consider randomly splitting up the original group of 100 cows into 2 groups.

Since each cow has a 50% chance of being brown, half of any subset of the cows should be expected to be brown.

If there is no bias, or selection process that affects the probabilities of the groups, then the groups are independent.

Here I've split the 100 cows into a group of 1 and a group of 99, and then changed the expected probability of the group of 1 with the information that the escaped cow is brown, but that doesn't affect the expected proportions of the other group.

 

 

 

It's an easier puzzle when you don't think about it! You can solve it many different ways, and the more complex solutions will work out the same as the simplest, due to the consistency of maths. By thinking about it, it becomes a brain teaser about intuitive (or unintuitive) aspects of stats, including the consequences of the independence of independent random variables, and also being convinced that they really are independent. A slight modification to the puzzle can change it, and make the variables not independent, which is probably a common source of confusion because when we try to think of real world examples, intuition of "how things should be" can sometimes add unnoticed biases.

[menageriemanor]

Just a little side note.

 

I believe there IS a blue breed of cow developed on King Island or some obscure Oz island, tho I believe some have white patches or flecking. I believe it is not yet officially recognised as a breed, but it's colour is to be a feature of the breed.

 

It was started from inbred cattle found on a farm on the island, and the farmer's wife loved the look of them, and when her husband sold cows, or a favorite bull, as mongrel, unimportant cattle, she'd storm over to the buyer and buy them back, for more, until he gave up.

 

.She kept blocking him from selling her best/favorite cattle, and then I think 40/50 years later, it was another woman, a newcomer to the island, who fell in love with them. At that point, they were kept on by the fanily, as mum's/grandma's blue herd, I think just on the strength of her personality. The new woman convinced them she wanted to keep and improve them and tracked down some blue coloured bulls on the mainland to outcross and improve type. She is now working on a breed standard and will eventually apply for breed recognition. I believe her steers are sold as organic rare beef, to some of the more expensive and trendy restaurants of Melbourne.

 

These cattle are blue, just as you can own a blue greyhound, which I did, as a pet, rescued from death row, for not really being bothered about racing. It's a blue/grey shade of the sort of colour you see in weimaraner, if you want to picture it. Officially, it IS blue.as recorded colour.

[N S]

If the supplier has randomly sold the cows with equal likelihood, then although the cow which escaped the barn was brown, it will not affect the habits of the supplier. The proportion of brown cows to total cows will be 50/100 or 1/2.

[LaurieAG]

Just a little side note.

 

I believe there IS a blue breed of cow developed on King Island or some obscure Oz island, tho I believe some have white patches or flecking. I believe it is not yet officially recognised as a breed, but it's colour is to be a feature of the breed.

 

It was started from inbred cattle found on a farm on the island, and the farmer's wife loved the look of them, and when her husband sold cows, or a favorite bull, as mongrel, unimportant cattle, she'd storm over to the buyer and buy them back, for more, until he gave up.

 

Sorry to disapoint you menageriemanor but King Island Blue is a cheese. The cows aren't blue that's the colour of the mold in the cheese.

 

http://kingislanddairy.com.au/

http://en.wikipedia.org/wiki/Blue_cheese

 

Save that one for April 1 LOL.

[menageriemanor]

LaurieAG I am well aware of King Island Blue Cheese. When it is on special and I feel outrageously decadent, I buy it. This is a cow herd whose main feature is the colour blue, JUST as the colour blue can be found in greyhounds. as their official colour, and I described it as best as I could, in writing.

 

When blue cheese graze in paddocks, mate, bleed, give birth, lactate, stampede to you, if you throw them apples, break legs so they have to be put down, then I'll consider taking your remark without outraged disbelief.

 

I live on acreage, own farm animals, and am very familiar with both cheese and cattle and can tell them apart. Do you seriously believe someone not you would be unable to tell a cow or bull from a cheese? Would they be able to construct a sentence?

 

I did say I THOUGHT it might be King Island OR another obscure island. Altho as an animal lover, I was vaguely interested, at the time, I didn't note the details of where to find the cattle or names of people or stud, as I'm almost vegetarian, and have no desire to own cattle, now or in the future, or to eat beef in Melbourne restaurants. Whatever colour the cattle.

 

As I recall there was only one 'purebred' herd, I would imagine there would not be a major promotion of the breed for eating, as one herd and first crosses culled, can hardly supply more than a couple of gourmet restaurants, year round.

 

I think, in a lifetime of all the usual sexist comments I've heard, being informed I don't know cattle from cheese, just about tops anything I can think of...

 

I suggest you try any or all islands between mainland Australia and Tasmania, given it is so important to you to prove I'm wrong. Perhaps it was Kangaroo Island. I'm pretty sure it was below the mainland. I don't remember where, but I saw live BLUE cattle, saw the woman who was breeding them, determined to retain the unusual colour, and know that she was marketing the organically raised steers to trendy Melbourne restautants. I also saw an interview with the grandson of the original farmer's wife. The chef interviewed on meat quality of blue cheese must have got a faulty batch? The steers are individually slaughtered and at some point, according to your superior biological knowledge, morph into blue cheese! That's how it's done! Who knew?

 

The only sad fact that might have ended the breeding programme, is there have been some appalling fires on some of those islands, in the last 2 years. There is a tiny chance, with only one herd, that they could have been lost. If a steer is burnt in a paddock, there is no bleeding way you are going to turn it into cheese.

[D H]

Apparently it's Norfolk Island, not King Island, where that blue breed of cattle live.

[menageriemanor]

I just went off to google it, myself. thanks DH.

 

LAG A point when starting to confirm a statement is to go with the definite fact BLUE CATTLE, not to google the possible area which was admitted to be doubtful. google Blue cattle and 5thish? down from the top of first page is the NORFOLK BLUE CATTLE. Since I saw the interviews, they have apparently started their own restaurant on the island, for their product. NORFOLK ISLAND is east of Australia, not south, part of NSW, officially, but the story before the blue cattle may have been from a southern island before, and I may have not realised it was a different story, when I finally noticed the blue cattle and started paying attention. The fact is I SUGGESTED it may have been King Island, which is very 'foodie' but I made it clear that was uncertain.

 

The reason they would have marketted to restaurants in Melbourne, is that is the 'foodie' capital of Oz, and amongst the fabulous food, are people who will jump on any thing that they can market exclusively, like, ridiculously, the colour of a cow, for steak.

 

There is already a BELGIAN BLUE cattle breed established which took the point I was making, that there ARE OFFICIALLY, blue cows, whether or not this woman is breeding cheese or cattle.

 

http://norfolkblue.com/about-norfolk-blue/

 

PS They do sell cheese BUT IT IS NOT MADE FROM THEIR BEEF and it is NOT specifically blue.

Edited February 18, 2013 by menageriemanor

[LaurieAG]

http://en.wikipedia.org/wiki/Australian_Cattle_Dog

 

 

The Australian Cattle Dog is a medium-sized, short-coated dog that occurs in two main colour forms. It has either brown or black hair distributed fairly evenly through a white coat, which gives the appearance of a "red" or "blue" dog. It has been nicknamed a "Red Heeler" or "Blue Heeler" on the basis of this colouring and its practice of moving reluctant cattle by nipping at their heels.

 

A mate had a Blue Cattle dog and crossed it with a Red Cattle Dog/Dingo cross and the result was a pure Brown Cattle Dog with no white.

[menageriemanor]

LAG I don't understand the point of the above. That 'blue' is not a dominant colour? or that adding a complete outcross in the preceding generation, ie the dingo, will dilute or lose the strong colour a breed is known for... Queen Anne's dead... 19th century British saying, that is really old news. (Also used by Brit history nerds and swots in 20th/21st century...)

 

Not cranky, not upset, give me a fortnight and I won't even remember who told me I didn't know cheese from live cattle. Or vice versa. Tho I have to say, normally I've forgotten whom I've argued with or verbally slapped down, in 2 days, and rarely, I spend a bit of time thinking, "Who am I feuding with?" but finding someone is willing to believe I could be that slack in my comments was a shock. I didn't chase confirmation, as it was just a casual, helpful contribution to urban science buffs who didn't know that BLUE could be an official colour, in animal breeds and species.

 

Anyway, I don't sulk, don't hold grudges, don't REMEMBER, unless someone seems to be callously uncaring about animals. Call your dog or cow, cheese. As long as it is happy and healthy, .I don't care.

[Consistency]

At the top of my head. 1 cow escapes.. you're left with 99 cows. 50 blue and 49 brown.

 

It wouldn't be 49 or 50. Maybe 49.5 brown to 50.5 blue?

Edited February 27, 2013 by Consistency