Respuesta :
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,
[tex]P(XXY)=0.8\times 0.9\times (1-0.2)[/tex]
[tex]P(XXY)=0.8\times 0.9\times 0.8[/tex]
[tex]P(XXY)=0.576[/tex]
[tex]P(XYX)=0.8\times (1-0.9)\times 0.2[/tex]
[tex]P(XYX)=0.8\times 0.1\times 0.2[/tex]
[tex]P(XYX)=0.016[/tex]
[tex]P(YXX)=(1-0.8)\times 0.9\times 0.2[/tex]
[tex]P(YXX)=0.2\times 0.9\times 0.2[/tex]
[tex]P(YXX)=0.036[/tex]
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
