Answer:
a: 0.0022
b: 0.00002
c: 0.0002
d: 924
e: 0.0025
f: 0.6652
g: Expected value 7.8 months, standard deviation 1.77 months
Step-by-step explanation:
This is a binomial situation. Either the market goes up, or it doesn't. (It could stay the same, but it doesn't offer that as an option, so we don't include it).
When calculating probability of a binomial situation, use the formula
(nCr)(p^r)(q^(n-r)) where p is the probability of success, q is the probabilty of failure, n is the the number of times the situation occurs, and r is the desired number of successful outcomes. The function nCr tells us how many ways you can choose r items from n total items. In this case, we are choosing some number of months, r, from 12 months, n
For our situation we have: n = 12, r = 12, p = 0.6, q = 0.4,
For a: (12C12)(0.6^12)(0.4^0) = (1)(0.0022)(1) = 0.0022
We want all 12 months of success and no months of failure
For b: (12C0)(0.6^0)(0.4^12) = (1)(1)(0.00002) = 0.00002
We want all months of failure, and no months of success
For c: (0.6^6)(0.4^6) = 0.0002
We don't need the nCr function because it tells us that it we want 6 months of success followed by 6 months of failure. There's only 1 way to do that, so there's nothing to count
For d: 12C6 = 924
There are 924 different ways to the market can go up 6 times out of 12 months
For e: (12C2)(0.6^2)(0.4^10) = (66)(0.36)(0.0001) = 0.0025
For f: Add the following probabilities together...
(12C7)(0.6^7)(q^5) + (12C8)(0.6^8)(q^4) + (12C9)(0.6^9)(q^3)
+ (12C10)(0.6^10)(q^2) + (12C11)(0.6^11)(q^1) + (12C12)(0.6^12)(q^0)
= 0.6652
They are mutually exclusive, the probability for the union is zero because they can't happen at the same time
For g: Find expected value with the formula E = np , here n = 12, p = 0.6, so
E = 12(0.6) = 7.8 months
Find the standard deviation with the formula s = √(npq), or s = √(Eq), here q = 0.4, so s = √(7.8(0.4)) = 1.77 months
