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If we have a carton of 12 eggs with 6 white and 6 brown then what is the probability that every egg is white?

There are two approaches that I can see. On the one hand we can say that the probability of the 6 white eggs being white is 1 and the probability of the 6 brown eggs being white is 0, and so the probability that every egg is white is 0. But on the other hand we can say that the probability of any given egg being white is 0.5 and so the probability of every egg being white is 0.5^12.

What's the correct approach to this?

2 Answers 2

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If we have a carton of 12 eggs with 6 white and 6 brown then what is the probability that every egg is white?

Zero. Because all the eggs aren't white, and there is no randomness anywhere.

The eggs are not all white

is a true statement.

But on the other hand we can say that the probability of any given egg being white is 0.5 and so the probability of every egg being white is 0.5^12.

No, the probability of a randomly selected egg being white is $0.5$. The randomness is introduced here via your selection of it. Without that, the probability of any given egg being white is either $0$ or $1$.

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    So to calculate the probability that every X is Y we don't take the probability of any given X being Y and raise it to the nth power, where n is the number of Xs? – 2017-02-02
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    I have no idea what that question is supposed to be asking. – 2017-02-02
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    Well, let's say I've already flipped three coins and asked you for the probability that all of them landed heads. You take the probability that one of them was heads (0.5) and raise it to the power of 3. – 2017-02-02
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    @MichaelRushton Yes, but there's *randomness* in what you are doing. You are flipping a coin, and there are two things that can happen. There is no randomness in the eggs simply being there and having some colour. A given egg is white, and there are no two things that can happen. The white egg is white. Unless, of course, you randomly select the egg. But you aren't doing that. – 2017-02-02
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The main reason why the argument that leads to probability $0.5^{12}$ is incorrect is that it is based on a formula (calculating the probability of event $A \land B$, as the product of probabilities of $A$ and $B$) that is only true when the events are independent.

Since a lot is known about the eggs (6 white and 6 brown), the 12 events "egg number $i$ is white" ($i=1,\ldots,12$) are not independent: If you known the value of 11 of them, for example, you also know the 12th.

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    Does this then mean that the probability is dependent on the known variables? For example, if I were to tell someone who doesn't know that there are 6 white and 6 brown eggs that the method used to pick them (from a box of 24 eggs, with 12 white and 12 brown) meant that there was a 0.5 chance of any given egg being white, would the 0.5^12 probability be correct? – 2017-02-02
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    *If* the method of picking the 12 eggs really had the chance of 0.5 for each egg, and each choice is independent from the others, then yes, the probability of each egg being white is $0.5^{12}$. But I'm a bit weary of your 24-box. A valid way to do what you say is to take a fair six sided die, throw it 12 times and for each throw, put a white egg from the 24-box into the 12-box for a result of 1-3 and a brown egg for a result of 4-6. What is *not* a valid way is to grab blindly into the 24-box and take one egg "at random" and do this 12 times (with the 24-box containing fewer eggs each time). – 2017-02-03