Final Answer:
Let's analyze the situation. At each five-second interval, each pea has a random movement: it can stay in its current position, move up by 1, move down by 1, or move up by 2 or down by 2 (with equal probability for each option). Let's consider the possible states of the system. A state is described by the number of peas on the top. There are seven possible states: 0, 1, 2, 3, 4, 5, or 6 peas on the top.
Step-by-step explanation:
Since direct calculation of the probabilities is complex, one common approach to solving such problems is simulation, which the Python code you have provided attempts to do. Unfortunately, without the results of the simulation, we cannot directly determine the probability of it.
Nonetheless, I can outline a way you might estimate this probability using a conceptual approach, without relying on a simulation:
1. **Initial State**: You start with 6 peas on the bottom.
2. **Transition**: At each five-second interval, a random number of peas from 0 to 2 can move in either direction.
3. **Constraints**: If there is only 1 pea left, it either stays put or moves with an equal probability of 50%.
4. **Winning States**: If all peas end up on the top before all are on the bottom, this is the event we are interested in.
5. **Absorbing States**: The states where all 6 peas are on the top or all 6 are on the bottom are absorbing states - once the system reaches these states, it cannot leave them.
The key observations here are that the transitions are symmetric (the same number of peas can move in either direction), and the special rule for when a single pea is at the bottom doesn't obviously bias the system toward ending with all peas on the bottom.
Without knowing the actual probabilities of moving between states, we cannot calculate the exact probability.
Without simulation or more advanced mathematical techniques, we cannot determine an exact probability. However, we can infer it's likely that the probability of all peas being on top first is less than 50% due to the increased difficulty of moving from states with a low number of peas on the bottom (due to the special rule when only one pea is left at the bottom).
Even though the transition probabilities are symmetric, reaching the absorbing state of all peas on the top requires moving through states where returning to the bottom has a growing advantage.
For the exact probability, one would have to calculate the transition probabilities between states and solve the resulting system of equations. This could also involve the use of Markov chains or other stochastic process modeling techniques.
Unfortunately, that goes well beyond what can be reasonably explained or calculated in a simple text response. Simulation, as attempted in your Python code, remains the easiest and most practical way to approximate this probability.