Session 2: Quantitative Methods: Basic Concepts Reading 8: Probability Concepts
LOS f, (Part 1): Calculate and interpret the joint probability of two events.
A parking lot has 100 red and blue cars in it.
- 40% of the cars are red.
- 70% of the red cars have radios.
- 80% of the blue cars have radios.
What is the probability of selecting a car at random and having it be red and have a radio?
Joint probability is the probability that both events, in this case a car being red and having a radio, happen at the same time. Joint probability is computed by multiplying the individual event probabilities together: P(red and radio) = (P(red)) × (P(radio)) = (0.4) × (0.7) = 0.28 or 28%.
|
Radio |
No Radio |
|
Red |
28 |
12 |
40 |
Blue |
48 |
12 |
60 |
|
76 |
24 |
100 |
What is the probability of selecting a car at random that is either red or has a radio?
The addition rule for probabilities is used to determine the probability of at least one event among two or more events occurring, in this case a car being red or having a radio. To use the addition rule, the probabilities of each individual event are added together, and, if the events are not mutually exclusive, the joint probability of both events occurring at the same time is subtracted out: P(red or radio) = P(red) + P(radio) ? P(red and radio) = 0.40 + 0.76 ? 0.28 = 0.88 or 88%.
What is the probability that the car is red given that you already know that it has a radio?
Given a set of prior probabilities for an event of interest, Bayes’ formula is used to update the probability of the event, in this case that the car we already know has a radio is red. Bayes’ formula says to divide the Probability of New Information given Event by the Unconditional Probability of New Information and multiply that result by the Prior Probability of the Event. In this case, P(red car has a radio) = 0.70 is divided by 0.76 (which is the Unconditional Probability of a car having a radio (40% are red of which 70% have radios) plus (60% are blue of which 80% have radios) or ((0.40) × (0.70)) + ((0.60) × (0.80)) = 0.76.) This result is then multiplied by the Prior Probability of a car being red, 0.40. The result is (0.70 / 0.76) × (0.40) = 0.37 or 37%.
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