Suppose I have 𝑁 coins arranged in a line. I can toss a coin up to 𝜂 times, but I’m only allowed to toss coins that are not adjacent to each other in the same round. Each toss is independent and has a success probability 𝑝.After each set of 𝜂 tosses, I count the number of successes and then select a new set of coins to toss (again non-adjacent).
I want to know how to calculate the number of such non-adjacent (“disjoint”) coins that can be tossed in each round, based on the number of successful tosses and the configuration.
Could someone help me formalize or model this process? An example would help:
Initial coins: $ C_1, C_2, \ldots, C_{10} $ (arranged in a line).
Round 1 : Toss every other coin: $ C_1, C_3, C_5, C_7, C_9 $.
Assume 3 coins are removed (expected value for $ \eta = 2, p = 0.3 $).
No. of coins removed =$1-(1-p^\eta) *$ no.of coins being tossed = $1-(1-0.3^2)*5$
Round 2: Re-select disjoint coins from the remaining 7, depending on which 3 were removed.
Case 1: Remove $ C_1, C_3, C_5 $
Remaining coins: $ C_2, C_4, C_6, C_7, C_8, C_9, C_{10} $
New maximal disjoint set:
Selecting every other coin from the left gives: $ C_2, C_4, C_6, C_8, C_{10} $ (5 coins)
However, $ C_6 $ and $ C_7 $ are adjacent and $ C_7 $ is still available.
Optimal selection: $ C_2, C_4, C_7, C_9 $ (4 coins)
Disjoint coins in Round 2: 4
Case 2: Remove $ C_3, C_5, C_9 $
Remaining coins: $ C_1, C_2, C_4, C_6, C_7, C_8, C_{10} $
New maximal disjoint set:
$ C_1 $ and $ C_2 $ are adjacent, so they cannot both be selected.
Choosing: $ C_1, C_4, C_6, C_8, C_{10} $ (5 coins)
But $ C_6 $ and $ C_7 $ are adjacent, and $ C_7 $ is available.
Optimal selection: $ C_1, C_4, C_7, C_{10} $ (4 coins)
Disjoint coins in Round 2: 4
Case 3: Remove $ C_5, C_7, C_9 $
Remaining coins: $ C_1, C_2, C_3, C_4, C_6, C_8, C_{10} $
New maximal disjoint set:
$ C_3 $ blocks $ C_2 $ and $ C_4 $, so choose other positions.
Selecting: $ C_1, C_3, C_6, C_8, C_{10} $ (5 coins)
Since $ C_7 $ is removed, there's no conflict with adjacency.
Optimal selection: $ C_1, C_3, C_6, C_8, C_{10} $ (5 coins)
Disjoint coins in Round 2: 5
Now I want to calculate the most likely number of batches k, each consisting of $\eta$ attempts required to get the success outcome on all the coins.
