The conditional probability that exactly k rolls resulted in a 3, given that at least one roll resulted in a 3, is of the form:
11−(c1/c2)c3(c3k)(1c2)k(c1c2)c3−k,for k=1,2,…,6.
Find the values of the constants c1, c2, and c3
To find the values of the constants c1, c2, and c3, we can make use of the given conditional probability formula and the information that it provides.
Let's analyze the given formula in parts:
1. The term (c1/c2) represents the probability of rolling a 3 on any given roll.
2. The term (c3^(c3k)) represents the probability of rolling exactly k 3's consecutively.
3. The term ((1-c2)^(k)) represents the probability of not rolling a 3 on the remaining (6-k) rolls.
4. The term (c1^c3-k) represents the probability of rolling a non-3 on the remaining (6-k) rolls.
Now, let's consider the information given in the formula:
- The formula is valid for k = 1,2,...,6, which means it should cover all possible values of k.
- The given formula should represent a valid probability distribution, which implies that the sum of probabilities for all possible values of k should add up to 1.
Using these observations, we can set up the equation:
∑[k=1 to 6] (11−(c1/c2)c3(c3^k)((1-c2)^k)(c1^c3-k)) = 1
Simplifying this equation will give us a system of equations involving c1, c2, and c3 that we can solve simultaneously.
Please note that without any additional information or constraints, it would be difficult to uniquely determine the values of c1, c2, and c3. To solve for these constants, we would need more specific information about the problem or restrictions on the values of c1, c2, and c3.