Conditional probability measures how strongly an event is believed when another event or condition is already known. The notation P(S|C) means the probability of sneezing given a cold; the vertical bar means “if,” not “or.” section 13.1
Relate conditional probability to the other parts of a probability model:
- Multiply a prior by a conditional probability to obtain a joint probability: P(C) × P(S|C) = P(C&S). section 13.1
- Represent conditional probabilities as likelihood functions that take a condition and return a distribution. section 13.6
- Use the joint distribution, filtering, and projection to recover a posterior probability such as P(C|S). section 13.3
Cold and sneezing
Among 1,000 people with colds, 860 sneezed, so P(S|C) is 86%. Among all 10,000 respondents, 860 both had colds and sneezed, giving P(C&S) = 8.6%. After observing a sneeze, the probability of a cold becomes P(C|S) = 860/1,130, or about 76.1%. section 13.1
In a discrete-probability program, a prior distribution and likelihood function can be combined into a joint distribution. Filtering for an observed effect and projecting the possible causes computes the posterior conditional distribution. section 13.3
Table 13.1 A survey of 10,000 respondents organized in a marginal table
| Cold: YES |
Cold: NO |
Total |
|
|---|---|---|---|
| Sneezed: YES |
860 |
270 |
1,130 |
| Sneezed: NO |
140 |
8,730 |
8,870 |
| Total |
1,000 |
9,000 |
10,000 |
Browse the marginal survey table to verify how conditional, joint, and marginal probabilities arise from the same counts. section 13.1
