Consider a collection of n biased coins, each showing Heads with probability p and Tails with probability 1-p , independently of the others. The coins are tossed and all coins showing Heads are collected together and tossed again. Write down an expression for the probability mass function of the random variable representing the number of Heads that turn up in the second round of tosses. Hint: Condition on the results of the first round of tosses using the “total probability" formula.

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To find the probability mass function (PMF) of the random variable representing the number of Heads that turn up in the second round of tosses, we can use the law of total probability.

Let's denote the random variable representing the number of Heads in the second round as X. We want to find P(X = k), the probability of getting k Heads in the second round.

We can consider the possible outcomes of the first round of tosses. Let's denote the random variable representing the number of Heads in the first round as Y.

Now, there are two cases to consider:

Case 1: In the first round, we get n Heads.
In this case, all the n biased coins are collected together and tossed again. The probability of getting a single Head in the second round is again p. Since there are n coins in the second round, the probability of getting k Heads in the second round, given n Heads in the first round, is given by the binomial distribution: P(X = k | Y = n) = (n choose k) * p^k * (1-p)^(n-k).

Case 2: In the first round, we get less than n Heads.
In this case, some of the biased coins do not turn up as Heads in the first round and will not be tossed again in the second round. Therefore, the number of coins that can potentially show Heads in the second round will be less than or equal to n. Let's denote this number as m, where m <= n. The probability of getting k Heads in the second round, given Y = m (where m < n), is again given by the binomial distribution: P(X = k | Y = m) = (m choose k) * p^k * (1-p)^(m-k).

Now, we need to consider the probability of each case happening. Let's denote the probability of getting Y = n as P(Y = n) and the probability of getting Y = m (where m < n) as P(Y = m).

Using the law of total probability, we can express P(X = k) as:

P(X = k) = P(X = k | Y = n) * P(Y = n) + P(X = k | Y = m) * P(Y = m)

Summing over all possible values of Y, we can generalize the expression as:

P(X = k) = Σ P(X = k | Y = m) * P(Y = m) for all possible values of m (from 0 to n).

This expression will give us the probability mass function (PMF) of the random variable X representing the number of Heads that turn up in the second round of tosses.