For each working day, in a local bank, there will be n people applying for personal loan. The probability that the
person’s application is successful is p. Given that the mean number of successful application is 27 and the variance
is 2.7.
(i) Calculate the values of n and p.
Using the values from part (b)(i),
(ii) find the probability that there were 25 successful application.
(iii) find the probability that the number of successful application is more than the mean value.
To solve this problem, we can use the concept of a binomial distribution.
(i) Let's start by calculating the values of n and p. The mean of a binomial distribution is given by the formula mean = n * p, and the variance is given by the formula variance = n * p * (1 - p).
From the given information, we know that the mean is 27 and the variance is 2.7. Therefore, we can set up the following equations:
27 = n * p (equation 1)
2.7 = n * p * (1 - p) (equation 2)
Now, we can solve these two equations simultaneously to find the values of n and p. However, since we don't have enough information to solve the equations explicitly, we will need to make some assumptions. Let's assume that n and p are positive integers.
From equation 1, we can see that n * p = 27. Since n and p are positive integers, let's try different combinations that multiply to 27:
1 * 27 = 27
3 * 9 = 27
9 * 3 = 27
27 * 1 = 27
Out of these combinations, let's check which satisfies equation 2 as well. Plugging in the values, we get:
For n = 1 and p = 27: variance = 1 * 27 * (1 - 27) = -702 (not equal to 2.7)
For n = 3 and p = 9: variance = 3 * 9 * (1 - 9) = -195 (not equal to 2.7)
For n = 9 and p = 3: variance = 9 * 3 * (1 - 3) = 54 (not equal to 2.7)
For n = 27 and p = 1: variance = 27 * 1 * (1 - 1) = 0 (not equal to 2.7)
None of the combinations satisfy both equations. This means that there is no integer solution for n and p that satisfy the given mean and variance. Therefore, we cannot determine the exact values of n and p.
(ii) Without knowing the exact values of n and p, we cannot directly calculate the probability that there were exactly 25 successful applications.
However, we can make an approximation using a normal distribution. The binomial distribution can be approximated by a normal distribution when n is large enough. Since we don't know the exact value of n, we cannot determine if it is large enough for this approximation to hold.
(iii) To find the probability that the number of successful applications is more than the mean value, we can use the concept of the cumulative distribution function (CDF).
P(X > 27) = 1 - P(X <= 27)
To calculate this probability, we need to know the exact values of n and p, which we have determined cannot be determined from the given information.
In conclusion, without the exact values of n and p, we cannot answer parts (ii) and (iii) of the question.