The Middlecreek Branch of the National Bank of Virginia has 10 real estate loans over $1,000,000. Of these 10 loans, 4 are "underwater." A loan is underwater if the amount of the loan is greater than the value of the property. The chief loan officer decided to randomly slect five of these loans to determine if they met all banking standards.
a. Waht is the probability that none of the selected loans is underwater?
b. What is the probability that 2 or more of the selected loans is underwater?
p under = .4
p not under = .6
p all five not under = .6^5
add p of 2 under + 3 under + 4 under + 5 under
or more easily 1 - (p 0 under + p 1 under)
p 0 under = .6^5 = .0778
p 1 under = C(5,1) * .4^1*.6^4 =.2592
so 1 - .2592 -.0778 = .663
To calculate the probability in this scenario, we need to use the concept of combinations and calculate the number of favorable outcomes and the total number of possible outcomes.
a. Probability that none of the selected loans is underwater:
To determine this probability, we need to calculate the number of combinations where all selected loans are not underwater and divide it by the total number of possible combinations for selecting any five loans.
First, let's calculate the number of combinations where none of the selected loans is underwater:
The number of loans not underwater = Total number of loans - Number of loans underwater
= 10 - 4
= 6
The number of combinations where none of the selected loans is underwater = Number of combinations of selecting 5 loans out of 6 non-underwater loans
= 6C5
= 6
Now, let's calculate the total number of possible combinations for selecting any five loans out of the ten loans available:
Total number of combinations for selecting 5 loans out of 10 = 10C5
= (10 * 9 * 8 * 7 * 6) / (5 * 4 * 3 * 2 * 1)
= 252
Therefore, the probability that none of the selected loans is underwater is:
Probability = Number of favorable outcomes / Total number of possible outcomes
= 6 / 252
= 1 / 42
So, the probability that none of the selected loans is underwater is 1/42.
b. Probability that 2 or more of the selected loans is underwater:
To determine this probability, we need to calculate the number of combinations where 2 or more selected loans are underwater and divide it by the total number of possible combinations for selecting any five loans.
First, let's calculate the number of combinations where 2 or more selected loans are underwater:
Number of combinations with 2 underwater loans = Number of combinations of selecting 2 loans out of 4 underwater loans
= 4C2
= (4 * 3) / (2 * 1)
= 6
Number of combinations with 3 underwater loans = Number of combinations of selecting 3 loans out of 4 underwater loans
= 4C3
= 4
Number of combinations with 4 underwater loans = Number of combinations of selecting 4 loans out of 4 underwater loans
= 4C4
= 1
Number of combinations where 2 or more of the selected loans are underwater = Number of combinations with 2 underwater loans + Number of combinations with 3 underwater loans + Number of combinations with 4 underwater loans
= 6 + 4 + 1
= 11
Total number of possible combinations for selecting any five loans out of the ten loans available is the same as calculated in part a, which is 252.
Therefore, the probability that 2 or more of the selected loans is underwater is:
Probability = Number of favorable outcomes / Total number of possible outcomes
= 11 / 252
So, the probability that 2 or more of the selected loans is underwater is 11/252.