Question

2 Suppose you are given $600 and decide to put it into a savings account. At the end of each year the amount either increases by $400 (with probability 0.75) or decreases by $200 (with probability 0.25). Suppose that if it ever reaches 0, then S200 is immediately paid into the account (so the next year starts with $200, not $0) (a) What is the expected time until there is $1,000 or more in the account? (b) What is the probability that $200 had to be paid into the account before it reached $1000?

Answer 1

Total Amount given = $600

Probability of amount increasing by $400 = 0.75

Probability of amount decreasing by $200 = 0.25

Expected increment at the end of the year = 0.75*400 - 0.25*200 = $250

Therefore, expected value at the end of the 1st year = 600 + 250 = $850

expected value at the end of the 2nd year = 850 + 250 = $1100

Therefore, at the end of the 2nd year , the balance would be $1000 or more

B. In order for the account to reach zero ( and be paid $200) before it becomes $1000, it needs 2 years., as that is the time taken for the account to become $1000.

This means the account balance has to reduce by $200 every year for the next 2 years and it has to become zero after 2years.

Probability that $200 will be deducted from the account in the next 2 years = 0.25*0.25 = 0.0625 = 6.25%

Hope I answered your query.