Answer:
The probability that the business makes a profit in exactly two of its first three years is 0.628.
Step-by-step explanation:
Given : Assume that the business makes a profit with probability 0.8 in the first year. For each year thereafter, the business makes a profit with probability 0.9 if it made a profit in the previous year, and with probability 0.2 if it did not make a profit in the previous year.
To find : What is the probability that the business makes a profit in exactly two of its first three years?
Solution :
Let X be the event that the business makes profit.
Y be the event that the business doesn't
.
The business makes a profit with probability 0.8 in the first year.
For each year thereafter, the business makes a profit with probability 0.9.
It did not make a profit in the previous year is 0.2.
According to question,
The business makes a profit in exactly two of its first three years which is given by, XXY, XYX, YXX
So,
![P(XXY)=0.8\times 0.9\times (1-0.2)](https://tex.z-dn.net/?f=P%28XXY%29%3D0.8%5Ctimes%200.9%5Ctimes%20%281-0.2%29)
![P(XXY)=0.8\times 0.9\times 0.8](https://tex.z-dn.net/?f=P%28XXY%29%3D0.8%5Ctimes%200.9%5Ctimes%200.8)
![P(XXY)=0.576](https://tex.z-dn.net/?f=P%28XXY%29%3D0.576)
![P(XYX)=0.8\times (1-0.9)\times 0.2](https://tex.z-dn.net/?f=P%28XYX%29%3D0.8%5Ctimes%20%281-0.9%29%5Ctimes%200.2)
![P(XYX)=0.8\times 0.1\times 0.2](https://tex.z-dn.net/?f=P%28XYX%29%3D0.8%5Ctimes%200.1%5Ctimes%200.2)
![P(XYX)=0.016](https://tex.z-dn.net/?f=P%28XYX%29%3D0.016)
![P(YXX)=(1-0.8)\times 0.9\times 0.2](https://tex.z-dn.net/?f=P%28YXX%29%3D%281-0.8%29%5Ctimes%200.9%5Ctimes%200.2)
![P(YXX)=0.2\times 0.9\times 0.2](https://tex.z-dn.net/?f=P%28YXX%29%3D0.2%5Ctimes%200.9%5Ctimes%200.2)
![P(YXX)=0.036](https://tex.z-dn.net/?f=P%28YXX%29%3D0.036)
The probability that the business makes a profit in exactly two of its first three years is given by,
P= P(XXY)+P(XYX)+P(YXX)
P= 0.576+0.016+0.036
P= 0.628
Therefore, The probability that the business makes a profit in exactly two of its first three years is 0.628.
= 0.272