Probability of a two-headed coin given a few sample flips?

Question

A probability problem was posed to me, and I am terrible at probability. It was first posed to me like this:

I have 5 coins. One of them is a two-headed coin, and the others are normal coins. I pick a coin, and then I flip that coin three times; each time is heads. What is the probability that the coin is two-headed?

My answer was, naturally, 1/5. I am still of the opinion that it's the correct answer to that particular phrasing (do you agree?). But of course I was told this is a wrong answer. After some thought, I came up with the following phrasing which is what was really meant:

You are given a coin and told that it has a 1/5 chance that it is double-headed, otherwise it is a normal coin. You are allowed three flips of the coin. Upon doing these, you receive three heads. What is the probability that your coin is double-headed?

Now I have two questions, first of all does that seem like a proper rewording to you? And secondly... What is the answer and the reasoning to reach that answer? I am terrible at probability, and while I can clearly see the problem and maybe spell out the first step or two, I am at a total loss as to how to arrive at an answer.

Edit: one more question, actually: is this an instance of the Monty Hall problem? I obviously see that it is very different, yet it feels somehow similar...

Answer 1

I think the rewording is fine, however, I believe your anwser is not correct.

Indeed, without throwing coins at all, there is a $1/5$ chance you get the double headed coin, and a $4/5$ chance you get a normal coin.

Now, the probability of getting three heads on three throws of a regular coin is $1/8$, as you can surely check. And as there is a $4/5$ chance you do get a regular coin, so you have a $1/8 \cdot 4/5 = 1/10$ chance you get three heads flipping a regular coin.

So to sum up the outcomes:

• $1/5$ chance of getting a double headed coin

• $1/10$ chance of getting a normal coin and three heads

• $7/10$ chance of getting a normal coin and anything but three heads.

However, note that you are asking: What is the probability that your coin is double-headed assuming you GET three heads? So the only options you are considering are the first two (as you know the third didn't occur).

Now there is a $1/5 + 1/10 = 3/10$ chance you get three heads. So there is a $\frac{1/5}{3/10} = 2/3$ chance of getting a double headed coin.

Answer 2

With the Bayesian interpretation of the question (your second phrasing), and leaving it as an exercise to figure out my notation: $$p(HHH|DH)=1$$ $$p(DH)=\frac{1}{5}$$ $$p(DH|HHH)=\frac{p(DH \wedge HHH)}{p(HHH|DH)p(DH)+p(HHH|N)p(N)}$$ $$=\frac{1/5}{1/5+(1/8)(4/5)}=\frac{2}{3}$$

Answer 3

Both your choice of the coin and the outcomes of the subsequent flips are random variables (though they are not independent). You want to calculate the probability of the coin choice given specific observed outcomes; i.e., you want a conditional probability. Here, you want $P(\text{two-head} \vert HHH) \equiv P(\text{two-head} \wedge HHH) / P(HHH)$. The numerator and the denominator are $$ P(\text{two-head} \wedge HHH) = P(\text{two-head})P(HHH \vert \text{two-head}) = \left(\frac{1}{5}\right)(1) = \frac{1}{5} $$ and $$ \begin{eqnarray} P(HHH) &=& P(\text{two-head} \wedge HHH) + P(\text{normal} \wedge HHH) \\ &=& P(\text{two-head})P(HHH \vert \text{two-head}) + P(\text{normal})P(HHH \vert \text{normal}) \\ &=& \left(\frac{1}{5}\right)(1) + \left(\frac{4}{5}\right)\left(\frac{1}{8}\right) \\ &=& \frac{1}{5} + \frac{1}{10} = \frac{3}{10}. \end{eqnarray} $$ The desired result is $$ P(\text{two-head} \vert HHH) = \frac{P(\text{two-head}\wedge HHH)}{P(HHH)} = \frac{1/5}{3/10} = \frac{2}{3}. $$

Answer 4

Urn models!

Tossing (Head,Head,Head) with one coin is like drawing a black ball out of an urn with 1 black and 7 white balls. The Head/Head coin is then like an urn with 8 black balls, let's imagine them as marked with a white dot.

Now, we put 5 urns (4 representing a normal coin and 1 representing the Head/Head coin) into a box. Because all urns contain the same number of balls, we can smash them without changing the probabilities: We can draw each of the 40 balls with same probability, and it will be with 1/5 probability from the urn representing the Head/Head coin.

If we draw any ball out of the box, that ball will be marked with the probability of 8/40, because 8 of 40 balls are marked. It will be black with 12/40. Under the precondition that we draw a black ball, it will be marked with the probability of 8/12.

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It is confusing, that we had the probability 1/5 before. So, what happened? How could we ignore the white balls?

• One way to ignore the white balls is this:

  We state, that we already know the case of tossing something else than (Head,Head,Head): 0 of 28 white balls are marked, so that means that we would know with the probability 28/40 that the coin has the probability 0/28 of being that Head/Head coin.

  That way we ask: "We know that none of the white balls are marked, but how many black balls are marked?"

• The other way lets us rephrase the question:

  You are given a coin and told that it has a 1/5 chance that it is double-headed, otherwise it is a normal coin. You are allowed three flips of the coin. Upon doing these, you receive either three heads or you give it back to restart this game. What is the probability that your coin is double-headed?

  That way we can ignore or remove the white balls out of the box.

Answer 5

Here's another way of thinking to get the correct result given above.

Say you take each of the coins (knowing whether it's H/H or H/T) in turn, and do 1000 runs of three flips.

For the four H/T coins the probability of a HHH result is $(\frac{1}{2})^3$ or 0.125, so you would expect to get 125 HHH runs from each of the four fair coins, or a total of 500 HHH runs.

For the one H/H coin, you would get, of course, 1000 HHH runs.

You'e now determined that $\frac{1000}{1500}$ or $\frac{2}{3}$ of the HHH runs will come from the H/H coin. Someone coming along and picking one particular HHH run from the whole experiment will find that $\frac{2}{3}$ of the time he was looking at a result from the H/H run.

Answer 6

This question is basically a puzzle to be solved with Bayesian statistics. But that might be for some already too much mathematics. Although this is a mathematics site, there may be some people that arrive on this question who can not read equations/math so easily. Therefore I thought that this answer might be helpful.

An extremer case:

The question may become more intuitive when you consider this problem:

• I have 5 coins. One of them is a two-headed coin, and the others are two-tailed coins. I pick a coin, and then I flip that coin three times; each time is heads. What is the probability that the coin is two-headed?

In this case you would not find it strange that the probability changes after observing how the coin is working. But now imagine that the two-tailed coins are false coins that flip 99% of the time on tails, should that change suddenly make such dramatic effect that we can not use experiments to become more certain about whether or not the coin is the double heads one?

Probability as frequency

Imagine you do the experiment many times. Then this is how often you get, on average, the results:

$$\begin{array}{ccll} \text{unfair coin} & 1/5 & \begin{cases} \text{unfair coin and }\hphantom{\text{not }}\text{3 heads in a row} & \\ \text{unfair coin and }\text{not 3 heads in a row} \end{cases} & \begin{array}{} 1/5 \times 1/1 &=& 2/10 \\ 1/5 \times 0/1 &=& 0/10 \end{array}\\ \text{fair coin} & 4/5 & \begin{cases} \hphantom{\text{un}}\text{fair coin and }\hphantom{\text{not }}\text{3 heads in a row} \\ \hphantom{\text{un}}\text{fair coin and }\text{not 3 heads in a row} \end{cases} & \begin{array}{} 4/5 \times 1/8 &=& 1/10 \\ 4/5 \times 7/8 &=& 7/10 \end{array}\\ \end{array}$$

So you see that observing the 'triple heads' occurs with a higher frequency for the false coin than for the fair coin. So one might put more believe (with probability = 2/3) in the case that one has landed in the situation with the false two-headed coin.

From an alternative viewpoint. For the cases that the coin is unfair, you will be always right. For the cases that the coin is fair, you will be only wrong for 1/8-th of the cases. You could say that the prior probability for drawing an unfair coin remains 1/5, but your response in the individual draws and after observations for specific cases is what differs from the 1/5-th frequency. The unfair coins occur with 1/5-th probability, but you will be able to filter out the unfair coins with 100% certainty and for the fair coins you will make mistakes only 12.5% of the time.

Small sidenote: you mention that this is related to the Monty Hall problem, and it is indeed related to the sticking to the initial selecting probability and rejecting/overlooking conditional probability, but it might be more related to the sleeping beauty paradox instead.