Consider a collection of biased coins, each showing Heads with probability and Tails with probability , 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.
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.
To find the probability mass function of the random variable representing the number of Heads that turn up in the second round of tosses, we can condition on the results of the first round of tosses using the "total probability" formula.
Let's assume we have n biased coins.
We can define a random variable X as the number of Heads in the second round of tosses. We want to find the probability mass function, P(X=k), where k represents the number of Heads in the second round.
We can express P(X=k) by conditioning on the results of the first round of tosses.
Let A_i be the event that i coins show Heads in the first round of tosses, for i = 0, 1, 2, ..., n.
Using the total probability formula, we have:
P(X=k) = P(X=k|A_0) * P(A_0) + P(X=k|A_1) * P(A_1) + ... + P(X=k|A_n) * P(A_n)
Now let's consider each term on the right-hand side.
P(X=k|A_0) represents the probability of getting exactly k Heads in the second round given that all coins showed Tails in the first round. Since all the coins showed Tails, there are no coins to toss in the second round. Therefore, P(X=k|A_0) = 0 for any k > 0, and P(X=0|A_0) = 1.
P(A_0) represents the probability that all n coins showed Tails in the first round. Since each coin shows Heads with probability p and Tails with probability 1-p, we have: P(A_0) = (1-p)^n.
For P(X=k|A_i) and P(A_i), where i ≠ 0, we need to consider the case when i coins show Heads in the first round.
In the second round, we have only i coins to toss. Each coin will show Heads with probability p, and Tails with probability 1-p. Therefore, the probability of getting exactly k Heads in the second round, given that i coins showed Heads in the first round, is given by the binomial distribution:
P(X=k|A_i) = (i choose k) * p^k * (1-p)^(i-k)
P(A_i) represents the probability that exactly i coins showed Heads in the first round. Since each coin shows Heads with probability p and Tails with probability 1-p, we have: P(A_i) = (n choose i) * p^i * (1-p)^(n-i)
Substituting these values into the total probability formula, we can obtain the expression for the probability mass function of X:
P(X=k) = 1 * (1-p)^n + (i choose k) * p^k * (1-p)^(i-k) * [(n choose i) * p^i * (1-p)^(n-i)]
Simplifying this expression further may depend on the specific values of n, p, and k.