Understand the problem: Understand the problem and the given probability function. $\newline$ We are given that the number of claims $N$ in a week follows a probability distribution $P[N=n] = \frac{1}{2^{(n+1)}}$ for $n \geq 0$ . We also know that the number of claims in one week is independent of the number in another week. We need to calculate the probability of receiving exactly $6$ claims over a two-week period.
Calculate one week probability: Calculate the probability of receiving exactly three claims in one week. $\newline$ Since the total number of claims we want over two weeks is $6$ , one way this can happen is if we receive three claims in the first week and three claims in the second week. Using the given probability function, we calculate this as $P[N=3] = \frac{1}{2^{3+1}} = \frac{1}{2^4} = \frac{1}{16}$ .
Calculate second week probability: Calculate the probability of receiving exactly three claims in the second week. $\newline$ Since the weeks are independent, the probability of receiving three claims in the second week is the same as in the first week, which is $P[N=3] = \frac{1}{16}$ .
Use independence to combine: Use the independence of the two weeks to calculate the combined probability. $\newline$ The probability of receiving three claims in the first week and three claims in the second week is the product of the two probabilities, because the weeks are independent. So, the combined probability is $(\frac{1}{16}) \times (\frac{1}{16}) = \frac{1}{256}$ .
Consider other combinations: Consider other combinations that result in $6$ claims over two weeks. $\newline$ We must consider that there are other ways to receive a total of $6$ claims over two weeks, such as receiving $2$ claims in the first week and $4$ in the second, or $4$ in the first and $2$ in the second, etc. We need to calculate the probabilities for all possible combinations that sum to $6$ .
Calculate probabilities for all combinations: Calculate the probabilities for all combinations that sum to $6$ . The possible combinations for $(n_1, n_2)$ where $n_1$ is the number of claims in the first week and $n_2$ is the number of claims in the second week, such that $n_1 + n_2 = 6$ , are $(0,6)$ , $(1,5)$ , $(2,4)$ , $(3,3)$ , $(4,2)$ , $(n_1, n_2)$ $0$ , and $(n_1, n_2)$ $1$ . We calculate the probabilities for each of these combinations using the given probability function.
Realize a mistake: Realize a mistake in the previous step. $\newline$ Upon reviewing the previous step, we realize that we made a mistake. We do not need to calculate the probabilities for all combinations because the question specifically asks for the probability of receiving exactly $6$ claims over the two-week period, not the distribution of those claims between the weeks. Therefore, we only need to consider the case where we receive three claims each week, which we have already calculated. We do not need to consider the other combinations.