Module 6: Probability and Probability Distributions
Module 6: Probability and Probability Distributions
Probability Rules (3 of 3)
Probability Rules (3 of 3)
Learning OUTCOMES
- Use conditional probability to identify independent events.
Independence and Conditional Probability
Recall that in the previous module, Relationships in Categorical Data with Intro to Probability, we introduced the idea of the conditional probability of an event.
Here are some examples:
- the probability that a randomly selected female college student is in the Health Science program: P(Health Science | female)
- P(a person is not a drug user given that the person had a positive test result) = P(not a drug user | positive test result)
Now we ask the question, How can we determine if two events are independent?
Example
Identifying Independent Events
Is enrollment in the Health Science program independent of whether a student is female? Or is there a relationship between these two events?
| Arts-Sci | Bus-Econ | Info Tech | Health Science | Graphics Design | Culinary Arts | Row Totals | |
| Female | 4,660 | 435 | 494 | 421 | 105 | 83 | 6,198 |
| Male | 4,334 | 490 | 564 | 223 | 97 | 94 | 5,802 |
| Column Totals | 8,994 | 925 | 1,058 | 644 | 202 | 177 | 12,000 |
To answer this question, we compare the probability that a randomly selected student is a Health Science major with the probability that a randomly selected female student is a Health Science major. If these two probabilities are the same (or very close), we say that the events are independent. In other words, independence means that being female does not affect the likelihood of enrollment in a Health Science program.
To answer this question, we compare:
- the unconditional probability: P(Health Sciences)
- the conditional probability: P(Health Sciences | female)
If these probabilities are equal (or at least close to equal), then we can conclude that enrollment in Health Sciences is independent of being a female. If the probabilities are substantially different, then we say the variables are dependent.
[latex]P(\text{Health Science}) = \frac{644}{12,000} \approx 0.054(\text{marginal probability; an unconditional probability})[/latex]
[latex]P(\text{Health Science | female}) = \frac{421}{6,198} \approx 0.068(\text{conditional probability})[/latex]
Both conditional and unconditional probabilities are small; however, 0.068 is relatively large compared to 0.054. The ratio of the two numbers is 0.068 / 0.054 = 1.25. So the conditional probability is 25% larger than the unconditional probability. It is much more likely that a randomly selected female student is in the Health Science program than that a randomly selected student, without regard for gender, is in the Health Science program. There is a large enough difference to suggest a relationship between being female and being enrolled in the Health Science program, so these events are dependent.
Comment:
To determine if enrollment in the Health Science program is independent of whether a student is female, we can also