Question

The accompanying table gives information on the type of coffee selected by someone purchasing a single cup at a particular airport kiosk.

	Small	Medium	Large
Regular	\(14\% \)	\(20\% \)	\(26\% \)
Decaf	\(20\% \)	\(10\% \)	\(10\% \)

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 he/she 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 small size was selected, and how does this compare to the corresponding unconditional probability of (a)?

Short answer

a. The probability of the given event S is \(0.34\) & for event D is \(0.4\).

b. The probability that he/she will use chosen decaf coffee is \(0.588\).

c. The probability that the small size was selected is \(0.5\).

Step-by-step solution

Definition of Probability

Probability denotes the possibility of something happening. It's a field of mathematics that studies the probability of a random event occurring. From \(0{\rm{ }}to{\rm{ }}1\), the value is expressed. Probability is a mathematical concept that predicts how likely occurrences are to occur. The definition of probability is the degree to which something is likely to occur.

 Calculation for determining the probability in part a.

We are given information on the type of coffee on the table in the exercise. We are selecting a random coffee purchaser.

Define event S - "the individual purchased a small cup". The probability of event S is

\(P(S)\mathop = \limits^{(1)} 0.14 + 0.2 = 0.34\)

(1): the sum of elements in column "Small".

Define event D - "the individual purchased a decaf coffee". The probability of event D is

\(P(D)\mathop = \limits^{(1)} 0.2 + 0.1 + 0.1 = 0.4\)

(1): the sum of elements in row "Decaf".

Calculation for determining the probability in part b.

We need to find the conditional probability of D given that the event S has occurred. Using the definition:

The conditional probability of A given that the event B has occurred, for which \(P(B) > 0\), is

\(P(A\mid B) = \frac{{P(A \cap B)}}{{P(B)}}\)

for any two events A and B.

The following is true

\(P(D\mid S) = \frac{{P(D \cap S)}}{{P(S)}}\mathop = \limits^{(1)} \frac{{0.2}}{{0.34}} = 0.588\)

(1): where the probability of the intersection of the events is given in the table ("Small" column, "Decaf" row), and the P(S) has been calculated in (a).

The interpretation is that \(58.8\% \) people who buy small coffee choose decaf.

Calculation for determining the probability in part c.

We need to find the conditional probability of S given that the event D has occurred. Using the definition, the following is true

\(P(S\mid D) = \frac{{P(S \cap D)}}{{P(D)}}\mathop = \limits^{(1)} \frac{{0.2}}{{0.4}} = 0.5,\)

(1): where the probability of the intersection of the events is given in the table ("Small" column, "Decaf" row), and the P(D) has been calculated in (a).

The interpretation is that \(50\% \)people who choose decaf buy small coffee.

Comparison: The conditional probability is bigger than the unconditional probability.