Jane Street

www.janestreet.com
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## Interview Question

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?

4

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
3

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
7

best answer: TT is counted as case 1 -> p=4/9 TH or HT is case 2 -> p=2/9 + 2/9 = 4/9 only discard HH Expected # of flips E=8/9*2 + 1/9*(2 + E) -> E = 9/4 or 2.25 flips

Anonymous on Feb 20, 2013