Final answer:
The probability of a path ending at 5 dollars, given the specified conditions of price movements and probabilities, is 0.384. This is calculated by identifying possible sequences of price movements that result in the desired final price, calculating the probability of each sequence, and summing these probabilities. The options provided in the question, however, do not match this calculated probability.
Step-by-step explanation:
To calculate the probability of a path ending at 5 dollars after starting at 4 dollars with 3 price movements, where each movement can be either up (1 dollar) or down (1 dollar), we need to find the sequences of movements (up or down) that result in a net increase of 1 dollar. Given the probabilities of up movements (0.2) and down movements (0.8), the only way to end at 5 dollars with 3 price movements is one up movement and two down movements (DUD, UDD, or DDU).
The probability of each sequence is calculated by multiplying the probabilities of each individual movement in the sequence. For example, for DUD it is (0.8 × 0.2 × 0.8). Since they are independent events, the overall probability for one sequence is the product of the individual probabilities.
There are 3 such permutations, and each has the probability of (0.8 × 0.2 × 0.8). To find the total probability of ending at 5 dollars, we add up the probabilities of each sequence:
Probability = 3 × (0.8 × 0.2 × 0.8) = 3 × 0.128 = 0.384
However, this result does not match any of the options given (A. 0.125, B. 0.625, C. 0.032, D. 0.128). It is important to recheck the computation or the understanding of the given options. If the error persists, it might be worthwhile to clarify the question with the student or consult additional resources.