*Not* sure what the answer is, ideas welcome!

You have 3 coins, one always comes up heads, the second always comes up tails, and the third is a fair coin. You select a coin at random. After selecting a coin, you flip it twice and get heads both times. What is the probability that your next flip is heads?

**Follow up:**
Suppose you flipped twice and got the *same* outcome both times (either HH or TT), then what's the probability that your next two flips match the first two (e.g. if you saw HH, your next two are HH)

I'm honestly unsure...

## Comments

Sign uporlog in with Facebookto comment.alvinfromvaThis would be more helpful with an answer!

^{ spoiler}I think it's .75.RaviWhat's your reasoning alvinfromva?

Jay ElliottAre you sure? How did you arrive at your answer?

GuilhermeSPOILERDON'T READ THIS BEFORE TRYING TO SOLVE THE PUZZLE************** * * * * * * * * * *SPOILERMy answer:

First case. Since you are going to get heads twice it necessarily means you did not choose the Tails coin. This gives you two scenarios:

1 - 1/2 chance of choosing the fair coin times 1/2 chance of getting heads again. 2 - 1/2 chance of choosing the Heads coin times 1 chance of getting heads again,

(1/2 * 1/2) + (1/2 * 1) = 3/4 or 75%

For the second case. Three scenarios:

1 - 1/3 chance of choosing the Heads coin times 1 chance of getting Heads twice. 2 - 1/3 chance of choosing the Tails coin times 1 chance of getting Tails twice. 3 - 1/3 chance of choosing the fair coin, but since you've already got the same result twice, you only have the probability of getting the same outcome twice again which would be 1/2 * 1/2 = 1/4.

1/3 * (1 + 1 + 1/4) = 3/4 or 75%

Did this pretty quick, not sure if I'm right.

RaviThe key is to infer the probability that each coin was selected based on the (heads,heads) outcome for the first two flips.

Label the three coins:

Data = HH. In general, as per Bayes' Rule, P(Ci | Data) is proportional to P(Data | Ci)*P(Ci), so:

Now, P(H | Data) = sum over Ci ( P(H | Ci) * P(Ci | Data ) = 0 * 0 + (1/2) * (1/5) + 1 * (4/5) =

9/10.In summary, probability of heads on the third flip (given the first two flips being HH) is 9/10.

Jay ElliottNice! Solid explanation.

BLRGHI computed it by computing the probability that the coin was the fair coin given that HH appeared, getting the probability that it was the doublehead coin from that, and then computing the probability of a head in those disjoint cases.

Just used Bayes's theorem and some arithmetic. The probability that you have a fair coin given the HH appearance is only 1/5; hence 4/5 it's the unfair heads coin (it's never the tails one obviously). The first comes from Bayes's theorem giving that, prob(fair coin | HH) is equal to P(HH|fair) P(fair) / P(HH), which is (1/4)(1/3)/(1/3 1 + 1/3 1/4) when keeping in mind the background info. Then (1/5)(1/2) + (4/5)(1) = 9/10 = 0.90

Confirmed it with a simple python program. http://codepad.org/yl4Be298

city_slickGood call applying Bayes' Theorem that way. My first thought was to use the HH information just to eliminate the all-tails coin. I think your logic is valid.

Sean MoranIf that's all true, which makes sense, so I'm not going to dispute it, then for the 2nd case it's the same for heads or tails as if you got 2 heads it's not the tails coin and if you got 2 tails it's not the heads coin so it's: Still 1/5 for the fair coin or 4/5 for the all-heads or all-tails coin then (1/5)(1/2)(1/2) + (4/5)(1)(1) = 1/20 + 4/5 = 1/20 + 16/20 = 17/20

AakashThat sounds right to me.

debashish ghatakI agree with ravi , the answer is 9/10.

Tarun245I think it is 25%

Asif Elahii could not understand