To calculate the marginal probabilities from a table of joint probabilities, you need to sum the joint probabilities across the appropriate dimensions.
For example, consider a joint probability table for two discrete random variables, A and B, represented as follows:
| P(A, B) | B1 | B2 | B3 |
|---|---|---|---|
| A1 | 0.1 | 0.2 | 0.1 |
| A2 | 0.2 | 0.3 | 0.1 |
| A3 | 0.1 | 0.1 | 0.1 |
To find the marginal probability of A, you sum the probabilities across all values of B:
- P(A1) = P(A1, B1) + P(A1, B2) + P(A1, B3) = 0.1 + 0.2 + 0.1 = 0.4
- P(A2) = P(A2, B1) + P(A2, B2) + P(A2, B3) = 0.2 + 0.3 + 0.1 = 0.6
- P(A3) = P(A3, B1) + P(A3, B2) + P(A3, B3) = 0.1 + 0.1 + 0.1 = 0.3
To find the marginal probability of B, you also sum the probabilities across all values of A:
- P(B1) = P(A1, B1) + P(A2, B1) + P(A3, B1) = 0.1 + 0.2 + 0.1 = 0.4
- P(B2) = P(A1, B2) + P(A2, B2) + P(A3, B2) = 0.2 + 0.3 + 0.1 = 0.6
- P(B3) = P(A1, B3) + P(A2, B3) + P(A3, B3) = 0.1 + 0.1 + 0.1 = 0.3
Thus, the marginal probabilities are:
- P(A1) = 0.4, P(A2) = 0.6, P(A3) = 0.3
- P(B1) = 0.4, P(B2) = 0.6, P(B3) = 0.3
These values represent the probabilities of each event occurring independently of the other variable.