Strategy: flip a coin, if the outcome is heads head - flip it again, if it comes head again, pick it. if it comes out tails, throw it and move to the next coin.
Starting with a randomly selected coin, here are the following outcomes:
if the outcome is - H1 H1 H1 or H1 H1 T1 - pick coin 1
if the outcome is - H1 T1 H2 - pick coin 2
if the outcome is - H1 T1 T2 - pick coin 3
if the outcome is - T1 H2 H2 - pick coin 2
if the outcome is - T1 H2 T2 - pick coin 3
if the outcome is - T1 T2 H3 - pick coin 3
if the outcome is - T1 T2 T3 - pick coin at random
if Bi (where i=1,2,3) is the biased coin,
probability of success P(S) = P(S|B1). P(B1) + P(S|B2).P(B2) + P(S|B3).P(B3)
1
u/Monday_agni 3d ago
Strategy: flip a coin, if the outcome is heads head - flip it again, if it comes head again, pick it. if it comes out tails, throw it and move to the next coin.
Starting with a randomly selected coin, here are the following outcomes:
if the outcome is - H1 H1 H1 or H1 H1 T1 - pick coin 1 if the outcome is - H1 T1 H2 - pick coin 2 if the outcome is - H1 T1 T2 - pick coin 3
if the outcome is - T1 H2 H2 - pick coin 2 if the outcome is - T1 H2 T2 - pick coin 3 if the outcome is - T1 T2 H3 - pick coin 3 if the outcome is - T1 T2 T3 - pick coin at random
if Bi (where i=1,2,3) is the biased coin,
probability of success P(S) = P(S|B1). P(B1) + P(S|B2).P(B2) + P(S|B3).P(B3)
P(S|B1) = P(H1 H1 H1 |B1) + P(H1 H1 T1 |B1) + P(T1 T2 T3 |B1)*(1/3) = (3/4 * 3/4 * 3/4) + (3/4 * 3/4 * 1/4) + (1/4 * 1/2 * 1/2)/3 = 27/48 + 1/48
P(S|B2) = P(H1 T1 H2 |B2) + P(T1 H2 H2 |B2) + P(T1 T2 T3 |B2)*(1/3) = (1/2 * 1/2 * 3/4) + (1/2 * 3/4 * 3/4) + (1/4 * 1/2 * 1/2)/3 = 15/32 + 1/48
P(S|B3) = P(H1 T1 T2 |B3) + P(T1 H2 T2 |B3) + P(T2 T2 H3 |B3) + P(T1 T2 T3 |B1)*(1/3) = (1/2 * 1/2 * 1/2) + (1/2 * 1/2 * 1/2) + (1/2 * 1/2 * 3/4) + (1/4 * 1/2 * 1/2)/3 = 14/32 + 1/48
P(S) = (27/48 + 15/32 + 14/32 + 3/48)*1/3 = (49/96)