Probability Rules Applied to an Everyday Decision: Addition, Multiplication and Conditional Probability Example

Discussion: Applying Probability Rules to an Everyday Decision (Chapter 3)

Chapter 3 introduces the mathematics of probability. In about 200–250 words:

• Choose a decision involving uncertainty that you or someone you know faced (umbrella, travel route, betting or medical diagnosis).
• Define two or more events (A, B) and calculate P(A or B) using the addition rule, P(A and B) using the multiplication rule when applicable, and P(A | B).
• State assumptions (independence, mutual exclusivity and so on) and reflect on how realistic they are.
• Based on your probabilities, what would you choose, and why?

Original example for study purposes (about 290 words). The probabilities are illustrative estimates; use figures from your own situation.

Decision. Each morning I decide whether to leave 20 minutes early for work so I am not late. Two uncertain events matter:

• A = it rains during my commute, P(A) = 0.30 (from the weather forecast).
• B = heavy traffic on the highway, P(B) = 0.40 (from my own records of the past few months).
• From my records, on rainy days traffic was heavy 60% of the time, so P(B | A) = 0.60.

Multiplication rule. P(A and B) = P(A) × P(B | A) = 0.30 × 0.60 = 0.18.

Addition rule. Rain and traffic can happen together, so they are not mutually exclusive: P(A or B) = P(A) + P(B) − P(A and B) = 0.30 + 0.40 − 0.18 = 0.52.

Conditional probability. P(A | B) = P(A and B) ÷ P(B) = 0.18 ÷ 0.40 = 0.45. Given heavy traffic, there is a 45% chance it is also raining.

Assumptions. If I had assumed rain and traffic were independent, I would have estimated P(A and B) = 0.30 × 0.40 = 0.12. That assumption is unrealistic, because rain slows drivers and causes accidents. Since P(B | A) = 0.60 is higher than P(B) = 0.40, the events are dependent. My estimates also assume that past months represent future conditions, which may not hold during holidays or road construction.

Choice. On any given day there is a 52% chance that at least one problem occurs, and when rain is forecast, the chance of heavy traffic rises to 60%. I would leave 20 minutes early whenever rain is forecast and consider leaving early on most other days, because the cost of being late is greater than the cost of a few extra minutes.

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