Module 6: Probability and Probability Distributions
Module 6: Probability and Probability Distributions
Probability Rules (2 of 3)
Probability Rules (2 of 3)
Learning OUTCOMES
- Reason from probability distributions, using probability rules, to answer probability questions.
Here we continue to use probability distributions to answer probability questions. We look for some patterns that suggest general rules for determining probabilities.
Example
When Can We Add Probabilities?
Compare these two questions. What do the solutions have in common?
- Question 1: A person with blood type A can receive blood from individuals with type A or O blood. What is the probability that a randomly selected person from the United States can donate blood to someone with type A blood?
| Blood Type | O | A | B | AB |
| Probability | 0.45 | 0.41 | 0.10 | 0.04 |
- Answer:P(donate to A) = P(blood type A or blood type O) = 0.45 + 0.41 = 0.86. There is an 86% chance that a randomly selected person in the United States can donate blood to someone with type A blood.
- Question 2:
What is the probability that a randomly chosen boreal owl nest will either be empty or contain only 1 egg?
| Number of Eggs | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| Probability | 0.2 | 0.1 | 0.1 | 0.25 | 0.25 | 0.05 | 0.05 |
- Answer:P(no eggs or 1 egg) = P(no egg) + P(1 egg) = 0.2 + 0.1 = 0.3. There is a 30% chance that a randomly selected boreal owl nest will be empty or contain only one egg.
What do these solutions have in common?
In each case, we have two events and we want to find the probability that either event A or event B occurs. In each case, we added the probabilities. This works because the events have no outcomes in common. When two events have no outcomes in common, they are disjoint.
The events “type A blood” and “type O blood” are disjoint. These events cannot both happen at the same time for a single person. A person cannot have both type A blood and type O blood.
The events “no eggs” and “1 egg” are disjoint. These outcomes cannot both happen at the same time for a single nest. A nest cannot contain no eggs and at the same time contain 1 egg.
If two events are disjoint, then we can add their individual probabilities. We write this fact as a rule:
P(A or B) = P(A) + P(B)
Comment
We stated the addition rule as a formal rule. A rule is a concise way to summarize a general principle from specific examples. This is one advantage of a rule. One disadvantage of a rule is that sometimes it discourages us from just thinking through a problem. Students often have the experience that they misremember a rule or forget the conditions required for the rule to work. This leads to mistakes that we can avoid if we just think through the problem without worrying about rules. We encourage you to think through probability problems whenever possible without resorting to rules. If you use a rule, be careful to check that the situation meets the conditions required for using the rule.
This addition rule for probabilities only works when the events are disjoint. If