Answer:
E(X) = 10 * 0.4 = 4
Step-by-step explanation:
Based on the information given, we can calculate the expected number of successful first-round appeals out of 10 as follows:
40% of the appeals are successful on the first round, which means that the probability of a successful appeal is 0.4.
The number of appeals out of 295,000 is not relevant for this calculation, as we are only interested in the proportion of first-round appeals that are successful.
Using these values, we can model the number of successful first-round appeals out of 10 as a binomial distribution with parameters n = 10 and p = 0.4. The probability of getting k successful appeals out of 10 can be calculated using the binomial probability mass function:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
where (n choose k) is the binomial coefficient, which can be calculated as n! / (k! * (n - k)!).
To find the probability of getting exactly k successful appeals out of 10, we can substitute the values of n, p, and k into the formula:
P(X = k) = (10 choose k) * 0.4^k * 0.6^(10 - k)
For each value of k from 0 to 10, we can calculate this probability and round the result to four decimal places:
P(X = 0) = (10 choose 0) * 0.4^0 * 0.6^10 = 0.0060
P(X = 1) = (10 choose 1) * 0.4^1 * 0.6^9 = 0.0403
P(X = 2) = (10 choose 2) * 0.4^2 * 0.6^8 = 0.1209
P(X = 3) = (10 choose 3) * 0.4^3 * 0.6^7 = 0.2144
P(X = 4) = (10 choose 4) * 0.4^4 * 0.6^6 = 0.2508
P(X = 5) = (10 choose 5) * 0.4^5 * 0.6^5 = 0.2007
P(X = 6) = (10 choose 6) * 0.4^6 * 0.6^4 = 0.1115
P(X = 7) = (10 choose 7) * 0.4^7 * 0.6^3 = 0.0425
P(X = 8) = (10 choose 8) * 0.4^8 * 0.6^2 = 0.0106
P(X = 9) = (10 choose 9) * 0.4^9 * 0.6^1 = 0.0015
P(X = 10) = (10 choose 10) * 0.4^10 * 0.6^0 = 0.0001
To find the expected number of successful first-round appeals out of 10, we can multiply each possible number of successful appeals by its probability, and then add up the results:
E(X) = 0 * P(X = 0) + 1 * P(X = 1) + 2 * P(X = 2) + ... + 10 * P(X = 10)
E(X) = 10 * 0.4 = 4