Answer: Please refer to Explanation
Explanation:
Firstly we have to remember that this is a Binomial Distribution.
We have p = 0.6 and you either have over $850 gross or you do not.
A binomial is described by (n choose k) p^k (1-p)^n-k
a) At least 3 out of 5 business days
So what this means is that we should find the probability when there are 3, 4, and 5business days greater than $850 and add them up.
Therefore we will have,
= (5 choose 3) .6^3 (.4)^2 + (5 choose 4) .6^4 (.4)^1 + (5 choose 5).6^5 (.4)^0
= 0.68256
b) At least 6 out of 10 business days.
Repeating the method above,
We will find p(6) + p(7) + p(8) +p(9) + p(10)
Which is,
= (10 choose 6) 0.6^6(0.4)^4 + (10 choose 7) 0.6^7(0.4)^3 + (10choose 8) 0.6^8(0.4)^2 + (10 choose 9) 0.6^9(0.4)^1 + (10 choose 10)0.6^10 (.4)^0
= 0.63310
c) fewer than 5 out of 10
From the previous question, we found at least 6 which meant we found probability when there are 6, 7, 8, 9, or 10 business days that will gross over $850.
Now, fewer than 5 means 4,3, 2, 1 , 0 business days grossing over 850) and seeing as we have already found at least 6 business days, we just need to find p(5) and then add it to p(at least 6) and subtract it from 1.
= (10 choose 5)(.6^5)(.4^5)
= .2006
p(at least 5) = p(at least 6) + p(5)
=0.63310 + 0.2006
= 0.83376
Subtracting from 1,
= 1-0.83376
= 0.1662
d) fewer than 6 out of the next 20 business days.
This is the same as finding p(0) + p(1) + p(2) + p(3) + p(4) +p(5) with n = 20
= (20 choose 0)(0.6^0)(0.4^20) + (20 choose 1)(0.6^1)(0.4^19) + ... +(20 choose 5)(0.6^5)(0.4^15)
= 0.0016
e) More than 17 out of the next 20 business days.
More than 17 means the same as p(18) + p(19) + p(20)
= (20 choose 18)(0.6^18)(0.4^2) + (20 choose 19)(0.6^19)(0.4) + (20choose 20)(0.6^20)(0.4^0)
= .0036
If you need any clarification do comment. Cheers.
