While you are waiting for your nets to fill, you play the following game: you have two coins, one is fair (heads/tails with equal probability), one is biased (heads twice as likely as tails). The coins are otherwise identical. You are going to choose a coin at random, flip it N many times, and then try to decide based on the results whether the coin is the fair coin or the biased coin. You win the game if you correctly identify the coin. (Your robot cat is able to distinguish which coin is which, and scores the game.)

You decide to follow the following strategy: if the majority of the flips are heads, you’ll guess it is the biased coin, otherwise you’ll guess that it is fair. That is, you’re setting a threshold of N/2 such that if you get that many heads or more, you guess it is the biased coin, otherwise you guess it is the fair coin.

1) Following this strategy, what’s the probability you win in a game of 10 flips? 20 flips?

2) If you want to maximize your probability of winning, what threshold should you set for your strategy in a game

of 10 flips? 20 flips? Be thorough as to your methods, and precise.

3) If the loaded coin had a probability p > 1/2 of giving H and 1 − p of giving T , where should you set your

threshold for your strategy, to maximize your probability of winning in a game of N flips?

4) Taking p as in the original problem before, if you could decide in advance how many flips you wanted before guessing - how many flips should you ask for if you wanted as few flips as possible, but still at least a 95% chance of winning? (Assume you use an optimal threshold for deciding if it is the fair or biased coin.)

Bonus 4: Imagine you have three coins: one is fair, one is biased (twice as likely to be heads as tails), the third is also biased (twice as likely to be tails as heads). You pick one of the three coins at random, and flip it 20 times. How should you decide which coin you are holding, based on the results, to maximize your probability of being correct? Be Thorough.

Hint:

All these problems really come down to understanding P(WIN). You can express P(WIN) as a function of the number of flips N, the threshold for decision making T, and the probability p for the heads-biased coin. You can say that P(WIN) = F(N,T,p), where F is some function of these three variables you need to figure out (it involves summation).

3.1 asks you to consider N = 10, T = 5, p as given.

3.2 asks you to consider N = 20, p as given, and find the value of T that makes F(20,T,p_given) as large as possible.

3.3 is more involved because now you don't have values for N or p, these are just variables. You want to find T in terms of N and p. So here's something to think about - for what values of T does increasing the threshold not help improve the probability of winning? i.e., when does F(N,T,p) >= F(N,T+1,p)? What can this inequality tell you (once you know what F is)?

Question

While you are waiting for your nets to fill, you play the following game: you have two coins, one is fair (heads/tails with equal probability), one is biased (heads twice as likely as tails). The coins are otherwise identical. You are going to choose a coin at random, flip it N many times, and then try to decide based on the results whether the coin is the fair coin or the biased coin. You win the game if you correctly identify the coin. (Your robot cat is able to distinguish which coin is which, and scores the game.)

You decide to follow the following strategy: if the majority of the flips are heads, you’ll guess it is the biased coin, otherwise you’ll guess that it is fair. That is, you’re setting a threshold of N/2 such that if you get that many heads or more, you guess it is the biased coin, otherwise you guess it is the fair coin.

1) Following this strategy, what’s the probability you win in a game of 10 flips? 20 flips?

2) If you want to maximize your probability of winning, what threshold should you set for your strategy in a game

of 10 flips? 20 flips? Be thorough as to your methods, and precise.

3) If the loaded coin had a probability p > 1/2 of giving H and 1 − p of giving T , where should you set your

threshold for your strategy, to maximize your probability of winning in a game of N flips?

4) Taking p as in the original problem before, if you could decide in advance how many flips you wanted before guessing - how many flips should you ask for if you wanted as few flips as possible, but still at least a 95% chance of winning? (Assume you use an optimal threshold for deciding if it is the fair or biased coin.)

Bonus 4: Imagine you have three coins: one is fair, one is biased (twice as likely to be heads as tails), the third is also biased (twice as likely to be tails as heads). You pick one of the three coins at random, and flip it 20 times. How should you decide which coin you are holding, based on the results, to maximize your probability of being correct? Be Thorough.

Hint:

All these problems really come down to understanding P(WIN). You can express P(WIN) as a function of the number of flips N, the threshold for decision making T, and the probability p for the heads-biased coin. You can say that P(WIN) = F(N,T,p), where F is some function of these three variables you need to figure out (it involves summation).

3.1 asks you to consider N = 10, T = 5, p as given.
3.2 asks you to consider N = 20, p as given, and find the value of T that makes F(20,T,p_given) as large as possible.
3.3 is more involved because now you don't have values for N or p, these are just variables. You want to find T in terms of N and p. So here's something to think about - for what values of T does increasing the threshold not help improve the probability of winning? i.e., when does F(N,T,p) >= F(N,T+1,p)? What can this inequality tell you (once you know what F is)?

Accepted Answer

Given The probability of heads or tails of fair coin =12=0.5The chances of heads is twice as…