The accompanying table gives information on the type of...

The accompanying table gives information on the type of coffee selected by someone purchasing a single cup at a particular airport kiosk. Consider randomly selecting such a coffee purchaser. (a) What is the probability that the individual purchased a small cup? A cup of decaf coffee? (b) If we learn that the selected individual purchased a small cup, what now is the probability that s/he chose decaf coffee, and how would you interpret this probability? (c) If we learn that the selected individual purchased decaf, what now is the probability that a small size was selected, and how does this compare to the corresponding unconditional probability from (a)?

The accompanying table gives information on the type of coffee selected by someone purchasing a single cup at a particular airport kiosk.
Consider randomly selecting such a coffee purchaser.
(a) What is the probability that the individual purchased a small cup? A cup of decaf coffee?
(b) If we learn that the selected individual purchased a small cup, what now is the probability that s/he chose decaf coffee, and how would you interpret this probability?
(c) If we learn that the selected individual purchased decaf, what now is the probability that a small size was selected, and how does this compare to the corresponding unconditional probability from (a)?

Key Concepts

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred. In problems like this, it involves recalculating probabilities based on new information, such as knowing that a small cup was selected or that decaf coffee was purchased. This is fundamental for updating our beliefs about the probability of one event when additional conditions are known.

Marginal Probability

Marginal probability refers to the probability of an event occurring regardless of the outcomes of other related events. In contexts like this problem, it is used to determine the overall chance of purchasing a small cup or decaf coffee without conditioning on any other specific information. It provides a 'baseline' probability before any additional conditional information is taken into account.

Joint Probability

Joint probability is the probability of two events occurring together, such as buying a small cup and decaf coffee at the same time. Understanding joint probabilities is crucial when determining the likelihood of combinations of outcomes, and it serves as the foundation for further computations of conditional probabilities.

Interpretation of Probabilities in Context

Interpreting probabilities involves understanding what the calculated numbers represent in real-world terms. For instance, when you determine the probability that a decaf coffee purchase came from a small cup, this value reflects how the knowledge of one attribute (cup size) affects the likelihood of another attribute (coffee type). This interpretation helps in understanding independent versus dependent events, and in analyzing how specific conditions can shift our expectations compared to the general trends represented by marginal probabilities.

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