An investment with initial value zero can go up or down $10,000 per month with equal likelihood. What is the probability that the investment will reach $1,000,000 on at least two different occasions, without reaching -$250,000 before it does so? Enter your answer as an exact fraction or as a number, rounded to 3 decimal places.

Step-by-Step Solution 1. Define Success and Failure Cases: Case 1: The investment fails to reach $1,000,000 at the first step (i.e., reaches -$250,000 1lE3R Case 2: The investment reaches $1,000,000 once but then fails to reach it a second time, either immediately or after some intermediate steps. 2. Probabilities for Each Case: e The probability of failing in the first step (Case 1) is g 3. Handling Case 2: * Once the investment reaches $1,000,000, we need to consider the probability of reaching it again before hitting -$250,000. The investment can be at either $1,010,000 or $990,000 after the first success. * The probability of failing in step 2 given we reached $1,000,000 once involves several transitions. The calculation simplifies as follows: P T P 250 P(Fail in step 2) = %[0]+% [1 124] = 1( . ) _ 5 4. Combining Probabilities: * The probability of not failing in step 2 and then failing in step 1 (i.e., reaching $1,000,000 once but not a second time): 1| ! P(Fail in step 2 and then failinstep 1) = - x = 250 1250 5. Total Failure Probability: * Combine the probabilities of Case 1 and Case 2: . 1 P(Case 1 or Case 2) = 3 e P 6. Final Probability: The probability of success (reaching $1,000,000 twice without hitting -$250,000): P(Success) = 1 - 1250 Simplifying: 1000 1 1001 1250 - 1001 249 P(Success) = 1 - + = 1 - = 1250 1250 1250 1250 1250