Consider a system of three molecules A, B, and C. Suppose that three units of energy can be distributed over the three atoms. Each atom can have no energy, one unit of energy, two units of energy, or all three units of energy. Assume that each of the ten possible arrangements of the three units of energy is equally probable. What is the probability that molecule C has some energy?
3/10
To find the probability that molecule C has some energy, we need to consider all the possible arrangements of the three units of energy and determine the number of arrangements where molecule C has at least one unit of energy.
Let's break down all the possible arrangements:
1. No energy in molecule C:
- This arrangement consists of all three units of energy distributed between molecules A and B.
- There are three atoms in total (A, B, and C) where the energy can be distributed, so we have C(3,3) = 1 arrangement for this case.
2. One unit of energy in molecule C:
- Out of the three units of energy, one unit can be allocated to molecule C and the remaining two units can be distributed between molecules A and B.
- There are three ways to choose the atom in molecule C to receive one unit of energy.
- For each of the above choices, there are two ways to distribute the remaining two units of energy in the other two molecules.
- Therefore, there are 3 * 2 = 6 arrangements for this case.
3. Two units of energy in molecule C:
- Out of the three units of energy, two units can be allocated to molecule C and one unit can be allocated to either molecule A or B.
- There are three ways to choose the two atoms in molecule C to receive two units of energy.
- For each of the above choices, there are two ways to choose which molecule (A or B) receives the remaining one unit of energy, and the other molecule receives no energy.
- Therefore, there are 3 * 2 = 6 arrangements for this case.
4. All three units of energy in molecule C:
- This arrangement consists of all three units of energy allocated to molecule C, and no energy is distributed to molecules A and B.
- There is only one arrangement for this case.
Now, let's calculate the total number of possible arrangements:
In the given scenario, there are ten possible arrangements of the three units of energy.
Adding up the arrangements from each case:
1 (no energy in molecule C) + 6 (one unit of energy in molecule C) + 6 (two units of energy in molecule C) + 1 (all three units of energy in molecule C) = 14 arrangements.
Therefore, the probability that molecule C has some energy is the number of arrangements where molecule C has some energy divided by the total number of arrangements:
Probability = (6 + 6 + 1) / 14 = 13 / 14.
Hence, the probability that molecule C has some energy is 13/14 or approximately 0.929.