Question

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. What is the probability that the business makes a profit in exactly two of its first three years?

Answer 1

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,

$P(XXY)=0.8\times 0.9\times (1-0.2)$

$P(XXY)=0.8\times 0.9\times 0.8$

$P(XXY)=0.576$

$P(XYX)=0.8\times (1-0.9)\times 0.2$

$P(XYX)=0.8\times 0.1\times 0.2$

$P(XYX)=0.016$

$P(YXX)=(1-0.8)\times 0.9\times 0.2$

$P(YXX)=0.2\times 0.9\times 0.2$

$P(YXX)=0.036$

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