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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.

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".

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.

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.