### Brain Teasers

# Stone-washed Genes

Probability
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

Both of my parents have brown eyes, as do I. My brother and my wife have blue eyes. Using the simple brown-blue model (two genes; a brown gene dominates blue gene), what are the chances of my first child having blue eyes?

### Hint

Given my brother's blue eyes, what are the odds on my pair of eye-color genes?### Answer

1 in 3.Since my brother has blue eyes (bb), both of my parents carry one brown and one blue gene (Bb). The three possibilities for my genotype, equally likely, are BB, Bb, and bB. Thus, there is a 2/3 chance that I carry a blue gene.

If I carry a blue gene, there is a 50% chance I will pass it on to my first child (and, obviously, 0% if I carry two brown genes).

Since my child will certainly get a blue gene from my wife, my gene will determine the eye color.

Multiplying the probabilities of those two independent events, there is a chance of 1/2 x 2/3 = 1/3 of my passing on a blue gene.

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## Comments

assuming that neither you or your wife have genes for a color that is recessive to blue.

I'm not an eye color expert, but is there any color, such as hazel or green, that is recessive to blue?

I'm not an eye color expert, but is there any color, such as hazel or green, that is recessive to blue?

TOO CONFUSING!

This is WRONG answer.

First, both parents were carriers- from proof via brother; there are only two genotypes the man got, Bb or BB; the "order" is meaningless and does not go into calculation

The correct answer is thus 1-from wife bb, 1 x 1/4; The man's genotype is either BB or Bb; there is thus a 1/4 chance he will pass on b gene.

john moyers, D.V.M.

First, both parents were carriers- from proof via brother; there are only two genotypes the man got, Bb or BB; the "order" is meaningless and does not go into calculation

The correct answer is thus 1-from wife bb, 1 x 1/4; The man's genotype is either BB or Bb; there is thus a 1/4 chance he will pass on b gene.

john moyers, D.V.M.

[quote]First, both parents were carriers- from proof via brother; there are only two genotypes the man got, Bb or BB; the "order" is meaningless and does not go into calculation[/quote]

Sorry, John, but the order is [i]not[/i] meaningless: it's the standard way of expressing the genes received from each parent. Yes, my genotype is either BB or Bb, but the two combinations (as opposed to permutations) are not equally likely.

Without loss of generality, let the left-hand position be the gene from my father, the right-hand one from my mother. Since they were both Bb genotypes (given that my borther has blue eyes), that means that there are four equally likely outcomes for any of their offspring: BB, Bb, bB, and bb.

Obviously, my brother is bb. However, that is the one combination that is not possible for my brown-eyed phenotype.

Therefore, the three permutations I could carry are BB, Bb, and bB, equally likely.

This leads to the given answer.

Sorry, John, but the order is [i]not[/i] meaningless: it's the standard way of expressing the genes received from each parent. Yes, my genotype is either BB or Bb, but the two combinations (as opposed to permutations) are not equally likely.

Without loss of generality, let the left-hand position be the gene from my father, the right-hand one from my mother. Since they were both Bb genotypes (given that my borther has blue eyes), that means that there are four equally likely outcomes for any of their offspring: BB, Bb, bB, and bb.

Obviously, my brother is bb. However, that is the one combination that is not possible for my brown-eyed phenotype.

Therefore, the three permutations I could carry are BB, Bb, and bB, equally likely.

This leads to the given answer.

H:

Little Brain has no comments.

Little Brain has no comments.

My first instinct was that the answer was wrong as well, but when I worked out all the possibilites by punnett square, it turns out that there is an 8/12 probability of brown eyes and 4/12 possibility of blue. good teaser!

norcekri, you tend to coment A LOT on you own teasers. could it be you're trying to compensate for somthing?

I agree with you smarty - and several others on site!

Maybe if you would put your explanations in words that most people could understand, then you would eliminate the need to go into such depth to explain your teasers!

Maybe if you would put your explanations in words that most people could understand, then you would eliminate the need to go into such depth to explain your teasers!

Excuse the double comment. But I would like to explain my last comment. Several of the members on site are young and will not be familiar with many of the words you use and will not understand what you are trying to explain. Maybe if you try to explain it in terms that everyone can understand, young and old alike, there will be less confusion as to what you are explaining. A less confusing explanation would probably prompt more people to leave comments as they would understand what you were trying to explain to them. Just a suggestion.

good one froggy! honestly, i didn't got it either

it doesnt matter....the answer is 1/4 not 1/3

the dude is either BB or Bb

make 2 punnet squares

in one its 50% brown eyes

in the other its 0%

that makes 1/4

Bb is the same as bB, you don't count both you moron

the dude is either BB or Bb

make 2 punnet squares

in one its 50% brown eyes

in the other its 0%

that makes 1/4

Bb is the same as bB, you don't count both you moron

I agree with the naysayers...1/4 is the correct answer, and I don't see how this teaser got accepted.

BB equals 0% chance of blue

Bb, or bB equals 50% chance of blue.

Thus (.5 x 0) + (.5 x .5) = .25

BB equals 0% chance of blue

Bb, or bB equals 50% chance of blue.

Thus (.5 x 0) + (.5 x .5) = .25

Nick, Viking, et alia,,

If you are trying to refute my solution, please make sure you refer to the problem as given. Simply asserting that I'm wrong doesn't solve the issue.

As I've pointed out in the solution and the later explanation, the BB and Bb cases are not equally likely. The 2/3-to-1/3 division comes when you properly apply Bayes' theorem to the given conditions. Each of you left out both the conditions and that application in your argument.

If you are trying to refute my solution, please make sure you refer to the problem as given. Simply asserting that I'm wrong doesn't solve the issue.

As I've pointed out in the solution and the later explanation, the BB and Bb cases are not equally likely. The 2/3-to-1/3 division comes when you properly apply Bayes' theorem to the given conditions. Each of you left out both the conditions and that application in your argument.

you are smoking something.

You got a big B, and a 50% of passing on that big B. Pass that big ole B on, any that baby of yours gets a big B...don't care if it's left or right handed. B be dominate so B it is.

You got a big B, and a 50% of passing on that big B. Pass that big ole B on, any that baby of yours gets a big B...don't care if it's left or right handed. B be dominate so B it is.

I really liked this teaser. Made me think for a couple minutes just to figure out what all the possibilities were. (And yes, your answer is correct.) Thanks!

norcekri is absolutely right. While everyone who is saying 1/4 instead of 1/3 is correct that the man is either BB or Bb, norcekri is right in the fact that Bb is twice as likely as BB. There is only one way that he could be BB (both his father and his mother gave him the dominant gene), while there are two ways he could be Bb (his father gave him the dominant gene and his mother the recessive, or vice-versa). Accounting for this, there is a 2/3 chance that his child will have a 50% chance of having blue eyes, and a 1/3 chance that his child will be guaranteed to have brown eyes, leading to a 1/3 chance that the child has blue eyes. Very good question about a confusing subject!

ok maybe i was wrong

Any chance I could get a biology credit out of this? Thanks for a swim in the gene pool ..I think the better part of wisdom is knowing what you don't know. I'm getting pretty wise on this site!

That was cool!

I was thinking of a follow-on teaser, and here is what I have:

Let's add a little more complexity...

The situation is the same -- two parents with brown eyes, a brother with blue eyes, a wife with blue eyes, and I have brown eyes. It turns out, though, that I already have two children, and they both have brown eyes. What is the probability that my NEXT child will have blue eyes?

[I do have an answer, but wanted to spark conversation first.]

Let's add a little more complexity...

The situation is the same -- two parents with brown eyes, a brother with blue eyes, a wife with blue eyes, and I have brown eyes. It turns out, though, that I already have two children, and they both have brown eyes. What is the probability that my NEXT child will have blue eyes?

[I do have an answer, but wanted to spark conversation first.]

Oct 21, 2005

I've got a different theory, which gives an answer of 3/8.

The odds are not 1/3 that the man has 2 Brown genes nor are they 1/2. The original possibilities from his parents were BB Bb bB and bb. So the probability that he inherited a BB is 1/4. This probability continues to be the case even though we have determined, after the fact by his actual eye color, that bb did not occur.

So... there is still just a 1/4 chance that he inherited BB. For the remaining 3/4 probability, representing the likelihood that he received at least one b gene from his parents, there is a 1/2 chance of sending a b (only 1/2 because we know the only possibilities are Bb and bB).

So the probability that the child will get a b from the man is 3/4 times 1/2 or 3/8.

note: Some may wonder why the chance of transmitting a b is only 1/2 when I demonstrated that you can't forget about bb as a possibility. The reason for this apparent anomaly is that even though his likelihood of having received at least one b is still 3/4, his odds of transmission to his child are affected by the knowledge that bb is not possible. In summary, while knowledge of results (his eye color) does not change his past probabilities, that knowledge will determine future probabilities affecting his child.

The odds are not 1/3 that the man has 2 Brown genes nor are they 1/2. The original possibilities from his parents were BB Bb bB and bb. So the probability that he inherited a BB is 1/4. This probability continues to be the case even though we have determined, after the fact by his actual eye color, that bb did not occur.

So... there is still just a 1/4 chance that he inherited BB. For the remaining 3/4 probability, representing the likelihood that he received at least one b gene from his parents, there is a 1/2 chance of sending a b (only 1/2 because we know the only possibilities are Bb and bB).

So the probability that the child will get a b from the man is 3/4 times 1/2 or 3/8.

note: Some may wonder why the chance of transmitting a b is only 1/2 when I demonstrated that you can't forget about bb as a possibility. The reason for this apparent anomaly is that even though his likelihood of having received at least one b is still 3/4, his odds of transmission to his child are affected by the knowledge that bb is not possible. In summary, while knowledge of results (his eye color) does not change his past probabilities, that knowledge will determine future probabilities affecting his child.

Zebediah -- I am not sure if you are kidding or not, so I will reply as if you were. If he is indeed still 1/4 to have BB, and 3/4 to have bB or Bb, and 0% to have bb (which is why he can't pass that on), you would be assuming that bB and Bb are equally likely at 3/8 apiece. So you would have information from the fact that he wasn't bb. If you instead say that he is 1/4 to be each of bb, Bb, and bB, then what is the other 1/4? It turns out that having additional information does let you limit the odds. As an example, I was equally likely to be conceived as xx (female) as xy (male). Given that I am now a father, what is the probability that I am xy? I am sure you will agree that it is 100%, in spite of the fact that there could have been another outcome initially.

Wouldn't it also depend on the genes of the woman you had the child with?

nice. (OMG I ONLY WROTE ONE WORD!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!)

LOL

LOL

Ok I skipped several of the last comments, so if I'm repeating anything, sorry. On a Punnet Square, the probability that the son (father) inherited BB is 1/4. Bb (the dominant gene is generally placed in front, order does not matter) =2/4. bb=1/4. he does not have bb. his wife does. if he inherited the BB gene, there is a 0% probability that his child will have blue eyes. however, if he got Bb, there is a 50% chance the child will be blue-eyed.

ok. now for the hard part. basically, he has 2 chances his gene is Bb, and only 1 that it's BB. so there's a 2:1 ratio that he will have the gene for his kids to be blue-eyed. Ok so there is 1/4 for him to have BB, so there is 1/4 chance that none of his kids are blue-eyed. But, there is a 2/4 chance he has Bb, and if he does his kid has a 2/4 chance to be blue-eyed. This takes into account the genes of his wife, who definitely has bb the recessive blue-eyed gene. So there is a 1/2 chance his kid will have blue eyes, and 1/4 that he will not.

Of course, all this is assuming that eye-color is as simple as a single gene. However, I for one don't want to think about all the possibilities THERE so it's safe to say that there's a 50% chance for a blue-eyed child, and a 75% chance (I think) for a brown-eyed one.

I'm fourteen, so obviously I haven't gotten into super-deep genetics yet. So I most definitely *do not* believe I can't be wrong. But I just wanted to explain what I thought made sense.

I liked this teaser, however. It was really challenging and made me think. Thanks.

ok. now for the hard part. basically, he has 2 chances his gene is Bb, and only 1 that it's BB. so there's a 2:1 ratio that he will have the gene for his kids to be blue-eyed. Ok so there is 1/4 for him to have BB, so there is 1/4 chance that none of his kids are blue-eyed. But, there is a 2/4 chance he has Bb, and if he does his kid has a 2/4 chance to be blue-eyed. This takes into account the genes of his wife, who definitely has bb the recessive blue-eyed gene. So there is a 1/2 chance his kid will have blue eyes, and 1/4 that he will not.

Of course, all this is assuming that eye-color is as simple as a single gene. However, I for one don't want to think about all the possibilities THERE so it's safe to say that there's a 50% chance for a blue-eyed child, and a 75% chance (I think) for a brown-eyed one.

I'm fourteen, so obviously I haven't gotten into super-deep genetics yet. So I most definitely *do not* believe I can't be wrong. But I just wanted to explain what I thought made sense.

I liked this teaser, however. It was really challenging and made me think. Thanks.

intresting, thanks

acurate but great. i liked it alot. Now scientsit are saying its not genetics, rather pigment related.anyone eles here this?

How many people on this site do not understand conditional probability. If I flip a coin and show you it is a head, you are stilli willing to bet it has a 50% chance of being a tail? Really? After the event has occurred and you know something about the outcome the probability changes. OF course he can't have 2 blue reccesive genes because he's got brown eyes! That leaves only 3 possibilties. The given answer is correct and quite well explained. Boo to the ad hominists who attack the person because they cannot understand conditional probability (albeit a difficult branch of Mathematics to master).

Good puzzle. Correct answer and well explained.

There is one possibility again. You can't just say that your parents (both) have Bb genes. Because it is still possible if one of your parents bring BB genes and the other Bb genes. It makes only two possibilities: Bb (brown) and bb (blue).

Because of that, the probability of your children is 50% for blue and 50% for brown.

Because of that, the probability of your children is 50% for blue and 50% for brown.

sorry, i'm mistaken. it is not BB but it is bb..

oops.. how can i be so stupid.. both BB and bb is wrong.. BB won't give your bro blu eyes. bb wrong too coz neither your parent has blu eyes. so the only one possibility left is your parents both have Bb genes.

did anyone do the punnett square method?

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