Jane Street Interview Question: There are 3 coins. One coin h... | Glassdoor

Interview Question

Quantitative Trader Interview Hong Kong (Hong Kong)

There are 3 coins. One coin has heads on both sides, one

  coin has tails on both sides, the third one has head on one side and tail on the other side. Now I pick up one coin and toss. I get head. What is the chance that the coin I picked has heads on both sides?

Interview Answer

13 Answers



Interview Candidate on Oct 17, 2010

i got this question too, but i am still not sure how it is calculated

still dun quite understand on Oct 19, 2010

why not 1/2?

Anonymous on Oct 19, 2010

Because you have 1/3 chance to get double head coin and you will surely get head, 1/3 chance to get single head coin and then 1/2 chance to get head. So the probability of choosing double head coin and get head is 1/3, while choosing single head coin and get head is 1/6. Then, given you get head after tossing, then chance that you chose double head coin is (1/3)/(1/3+1/6) = 2/3

Anonymous on Oct 19, 2010

That can't be right. We know that the chosen coin landed heads up, so we can disregard the third (Tails/Tails) coin. Of the remaining 2 coins, there's a 1/2 chance that the chosen one is the double-headed coin. The answer should be 1/2, no?

Are you sure? on Nov 15, 2010

2/3 is correct... review http://en.wikipedia.org/wiki/Bayesian_inference

Yellow on Nov 17, 2010

Let HH, TT, HT be the events defined by picking the head-head, tail-tail, and head-tail coins, respectively. Given that we pick one of the coins randomly, we obtain
P(HH)=P(TT)=P(HT)=1/3. Let H and T be the head and tail events respectively. The probability of event H given HH is clearly 1. That is, P(H|HH)=1. Furthermore,
P(H|HT)=1/2 and P(H|TT)=0. The probability of a heads is:
P(H)=P(H int HT)+ P(H int HH)+P(H int TT)=
P(H|HT)P(HT)+ P(H|HH)P(HH)+P(H|TT)P(TT)=(1/2)(1/3)+ (1)(1/3)+(0)(1/3)=
Note that P(H int HH)=P(H|HH)P(HH)= 1/3.
Therefore, the probability of event HH given H is:
P(HH|H)=P(H int HH)/P(HH)= (1/3)/ ((1/3)(3/2))=2/3.

Interview on Jan 2, 2011

2 heads on double headed coin, 1 head on the other, P(head is coming from double headed) = 2/3

jkramer on Jan 28, 2011

conditional probability?
probability the double headed given a head = probability (head unoin probability double headed coin)/ probability head is thrown.

= 1/3 / 1/2 = 2/3

anom on Jan 29, 2011

most of you are missing a fact that can't be ignored. Read the question - is says if you draw heads, meaning only 2 coins can have heads, what are the odds that you drew the 2 headed coin is 50%!

prm on Jun 21, 2011

C1 := 2 tail coin
C2 := 1 head 1 tail coin
C3 := 2 head coin

P(C3|H)=P(H|C3).P(C3)/P(H) = 1.(1/3)/(1/2) = 2/3
P(C2|H)= P(H|C2).P(C2)/P(H) = (1/2).(1/3)/(1/2) = 1/3

Anonymous on Nov 21, 2011

This is kind of like the Monty Hall Problem so check that out too.

Easiest way to see that it is definitely not 1/2 is to think of the same problem with more possible out comes than just heads or tails:

Imagine there are 101 boxes each with 100 balls in The first box has all blue balls, the next all red... etc with different colours in each box until the 100th which is all yellow say. The 101st has 1 ball of each colour. Suppose you randomly pick one box and pick one ball out at random.

The ball is blue.

Which is more likely?

That you happened to pick the 1 multi coloured box out of the 101 boxes and picked the one ball in that box that was blue.


That you picked one of the 100 boxes that were all one colour and that colour happened to be blue.

Hopefully you can see that these are not equally likely events.

Not 1/2 on Mar 17, 2012

2/3 : this is an application of Bayes theorem

Roland on Oct 14, 2015

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