Jane Street Interview Question: Unfair coin with P(H) = 1/3 a... | Glassdoor

## Interview Question

Quant Interview New York, NY

# Unfair coin with P(H) = 1/3 and P(T) = 2/3. a) How to make

an event with 50% probability? b) Expected number of flips until a realization occurs? c) Can you create a strategy to reduce the number of flips necessary? d) Can you create a strategy to reduce the number of flips necessary for an unfair coin with any bias?

5

a) Event 1 = {T,H}, Event 2 = {H,T}, if any other outcome, then re-roll
b) The probability of Event 1 or Event 2 occurring is 1/9+1/9=2/9. The expected number of 2-roll "tries" is 9/2. And each "try" consists of two rolls so 9 expected rolls for a realization.
c) Event 1 = {T,T}, Event 2 = any other combination. Probability of either event is 4/9.
d) Many solutions. Trick is to not discard any rolls. Use strategy from part (a), but if you roll {T,T}, then continue and put {T,T,H,H} in Event 1 and {H,H,T,T} in Event 2.

Interview Candidate on Apr 1, 2012
0

the c) is wrong. probability of event 2 is 5/9 not 4/9

mokhlos on Jul 1, 2012
1

Interview Candidate on Jul 1, 2012
5

Interview Candidate,

P(H) = 1/3 and P(T) = 2/3 so a H-T combo will have a 2/9 prob.
For a), the probability of Event 1 or Event 2 occurring is 2/9 + 2/9 = 4/9.
Therefore, the expected number of 2-roll "tries" is 9/4.
Each try consists of two rolls - so an expected 9/2 rolls until a realization.

Neil on Jan 3, 2013
10

TT is counted as case 1 -> p = 4/9
TH or HT is case 2 -> p = 2/9 + 2/9 = 4/9

Expected # of flips E = 8/9*2 + 1/9*(2 + E) -> E = 9/4 or 2.25 flips

Anonymous on Feb 20, 2013
0

What is the best (minimum number of expected tosses) possible for any strategy ?

Sinha on Mar 27, 2017
0

Using Information Theory, Expected number of coin-tosses required is at least (1/ H(1/3))=1.08

Anon on Mar 27, 2017