The chance of picking the biased coin is 1/1000. The chance of seeing 10 heads from a fair coin is (1/2)^10 = 1/1024. These are nearly equal, so given that you've seen 10 heads, there is a 50/50 chance of having a biased coin. So the probability the next flip shows a head is
You didn't read it wrong, but you probably did fail the test ;-) There is no gotcha in the question, it's just a math problem that you either do or do not know how to solve. This isn't really about intelligence as much as it is about whether you have taken a course on probability. If you flipped 10 heads in a row the probability of the coin you have being the double heads coin increases dramatically, so you have to take that into account for the next flip. For intuitive understanding it often helps to go to extremes. Suppose you do 1 billion flips and all come up heads. What is the probability that the next flip comes up heads? Because we had 1 billion heads it is virtually certain that we are dealing with the double heads coin, so the probability that the next flip will come up heads is close to 1.
"If you flipped 10 heads in a row the probability of the coin you have being the double heads coin increases dramatically..."
I disagree. The coin is the coin. It didn't magically transport itself or change state after flipping it 10, 100, or a billion times.
Lets change the puzzle to the simplest state: you pull a coin from the 1000-coin jar and flip it just once. What's the probability of heads?
This is why roulette and baccarat tables in Vegas have those signs showing previous outcomes. It's meant to mess with your head. Previous history has no effect on future outcomes. A fair coin could come up heads a billion times in a row as well. The next flip will still be 0.5.
The coin is the coin, but your information about the coin has changed. Probability is a fact about you, not a fact about the coin.
Here is perhaps a more visceral example: On any given day, your car has a 10% chance of having blown a wheel the previous night. If it has blown a wheel, every bump will feel jarring. If it hasn't blown a wheel, any given bump will feel jarring with 10% probability. You drive over a hundred bumps, and all feel jarring. Do you think the next bump will also feel jarring?
Yes, right? Because the fact that the last hundred bumps have been jarring means you probably blew a wheel last night. Even though 'previous history has no effect on future outcomes', previous history does give you information about the state of the world.
Edit: Or a better example, for this community. You're on a slightly flakey wifi connection, which sometimes drops a packet at random. Any given packet is dropped with probability 1/100. Also, any given time you connect to the router, the modem is down with probability 1/10, and all of your packets will get dropped. You connect, and the first hundred packets you send are dropped. What is the probability the next packet will get dropped? Very high, because now you know that the modem is probably down. The history of the connection gives you information about the state of the connection. Similarly, the history of coin toss results gives you information about the state of the coin.
(Why is this different from roulette? Because there the previous history doesn't give you information.)
So, I think that tge question is ambiguous, and a little more direction could clear it up. If you want to know the probability, taking into account the 10 heads results thus far, I would tell the interviewee, "I am going to do this process repeatedly (picking a coin and flipping 10 times), and I'm going to keep track of what percentage do the all one side thing. What will that percentage be approximately?".
Probability is a fact about you, not a fact about the coin.
Huh? It's a coin. You flip it. It has no knowledge of the past and no idea about the future. It lands and it's either heads 50% of the time and tails 50% of the time. It's all about the coin and nothing about what you've observed in the last N trials.
That's the simple logic that the OP is trying to see if you understand.
I think you're incorrect because your conceptual model of what constitutes "probability" is incorrect for this type of problem.
Try thinking about it in a more brute force way: imagine literally all possible outcomes of performing this experiment. In other words, create a list like this (each coin in the jar is numbered from 000 to 999 with 999 being the only biased coin, and coin flips are represented by 0 being heads and 1 being tails):
Now select all of the lines above where the first ten flips are heads. Of these outcomes, how many have an eleventh flip of heads and how many have an eleventh flip of tails? Unless my idea of probability is flawed, this should be the same answer that the mathematicians in this thread are providing, so something right around 75% heads.
Where is the unfair coin in your list? Without that you just get 50% heads. This method can definitely work to get the correct answer, but you have to account for all possibilities and weigh them by their probability.
The important thing is that as you observe heads from the coin, you learn something about the coin. As you observe more heads it is less likely to be a fair coin and more likely to be the coin with double heads. This doesn't change anything about the coin, but it changes something about what you know about the coin. See here for the correct answer: https://news.ycombinator.com/item?id=7000523
I think you're missing the point of the question which seems to get at whether the interviewee is familiar with/understands bayesian probability. For an excellent explanation you should see: http://yudkowsky.net/rational/bayes.
I don't see why knowledge of Bayesian probability is necessary for this question. I don't have anything more than a passing knowledge of Bayesian probability, but I was able to produce the correct answer and a reasonable explanation using what I believe to be classical probability: https://news.ycombinator.com/item?id=7001288
>>>Previous history has no effect on future outcomes. A fair coin could come up heads a billion times in a row as well. The next flip will still be 0.5.
You hit the nail on the head here - a fair coin would. But if there are unfair coins, things start to change. If we know there's an unfair coin, and we see an unusual run, we should consider that it may be that coin.
Think of this: increase the flips to 1000, and you're still getting heads each time. Would you bet on the next being heads or tails?
The probability of which coin you have affects the probability of the next toss coming up heads, so having this knowledge is implicit in determining the solution.
That probability was determined the moment you picked the coin out of the jar.
It makes no difference what you do to it after the pick. Hold it in your hand for a day, flip it 10 times, sit on it, whatever - the end result is that p(heads) for that particular coin has not changed. p(heads) will be either 0.5 for a real coin or 1.0 for the rigged one.
The probability then comes down to what coin you picked at the start of the trial. There's a 0.999 chance you have a real one, and 0.001 chance that you have the rigged one.
When we pick the coin, we have 1 in 1000 chance of getting the double heads coin, and 999 in 1000 chance of getting a fair coin. Lets call this P(fair) = 0.001, and P(fake) = 0.999.
When we have the double heads coin, the probability of getting 10 heads is 1: P(10 heads|fake) = 1. When we have a normal coin, the probability of getting 10 heads is P(10 heads|fair) = 0.5^10.
The chance of picking the biased coin is 1/1000. The chance of seeing 10 heads from a fair coin is (1/2)^10 = 1/1024. These are nearly equal, so given that you've seen 10 heads, there is a 50/50 chance of having a biased coin. So the probability the next flip shows a head is
The long answer -Yo want to figure out P(biased | 10H). Using Bayes rule this is
and you now compute the probability of the next toss being a head as above.