Assume that the business makes a profit with probability 0.4 in the first year. For each year thereafter, the business makes a profit with probability 0.6 if it made a profit in the previous year, and with probability 0.4 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?

Question

Assume that the business makes a profit with probability 0.4 in the first year. For each year thereafter, the business makes a profit with probability 0.6 if it made a profit in the previous year, and with probability 0.4 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?

Troy Lesure · Accepted Answer

Step 1   Introduction:   If the business makes a profit in exactly two of its first three years, then it may be in the first and second years, or the first and third years, or second and third years.   Step 2   Calculation:   First case:   Suppose the business makes a profit in the first and second years.   The probability that it makes a profit in the first year is 0.4.   Given that the business made a profit in the first year, the probability that it makes a profit in the immediately next or second year is 0.6.   As the business is expected to make a profit in exactly two of its first three years, and it has already made a profit in the second year, we are interested in the situation where it will not make a profit in the third year. Now, the probability that the business makes a profit in any year given that it made a profit in the previous year is 0.6. As a result, the probability of the complement of this event, that is, the probability that the business does not make a profit in any year given that it made a profit in the previous year is 0.4 [= 1 – 0.6]. Thus, the probability that the business does not make a profit in the third year, when it makes a profit in the second year, is 0.4.   Thus, the probability that the business makes a profit in its first and second years, and not on its third year, is:   (0.4)⋅(0.6)⋅(0.4)=0.096.   Second case:   Suppose the business makes a profit in the first and third years.   The probability that it makes a profit in the first year is 0.4.   Given that the business made a profit in the first year, the probability that it does not make a profit in the second year is 0.4.   Given that the business does not make a profit in the second year, the probability that it makes a profit in the immediately next or third year is 0.4.   Thus, the probability that the business makes a profit in its first and third years, and not on its second year, is:   (0.4)⋅(0.4)⋅(0.4)=0.064.   Third case:   Suppose the business makes a profit in the second and third years.   The probability that it makes a profit in the first year is 0.4, so that the probability of the complement, that it does not make a profit in its first year is 0.6[=1−0.4].   Given that the business does not make a profit in the first year, the probability that it t makes a profit in the second year is 0.4.   Given that the business makes a profit in the second year, the probability that it makes a profit in the immediately next or third year is 0.6.   Thus, the probability that the business makes a profit in its first and third years, and not on its second year, is:   (0.6)⋅(0.4)⋅(0.6)=0.144.   Thus, the probability that the business makes a profit in exactly two of its first three years is 0.304[=0.096+0.064+0.144].

Assume that the business makes a profit with probability 0.4 in the fi

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