find the probability that a drink is a coffee, given it is a small.\n| |large|small|total|\n|--|--|--|--|\n|c…

find the probability that a drink is a coffee, given it is a small.\n| |large|small|total|\n|--|--|--|--|\n|coffee|8|7|15|\n|tea|5|13|18|\n|total|13|20|33|\np(small and coffee) = \\frac{7}{33} p(small) = \\frac{20}{33}\ndivide and simplify.\np(coffee | small) = \\frac{p(small and coffee)}{p(small)} = ?

find the probability that a drink is a coffee, given it is a small.\n| |large|small|total|\n|--|--|--|--|\n|coffee|8|7|15|\n|tea|5|13|18|\n|total|13|20|33|\np(small and coffee) = \\frac{7}{33} p(small) = \\frac{20}{33}\ndivide and simplify.\np(coffee | small) = \\frac{p(small and coffee)}{p(small)} = ?

Answer

Explanation:

Step1: Recall conditional - probability formula

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. Here, $A$ is the event that the drink is coffee and $B$ is the event that the drink is small. We are given $P(\text{small and coffee})=\frac{7}{33}$ and $P(\text{small})=\frac{20}{33}$.

Step2: Substitute values into formula

$P(\text{coffee}|\text{small})=\frac{P(\text{small and coffee})}{P(\text{small})}=\frac{\frac{7}{33}}{\frac{20}{33}}$.

Step3: Simplify the fraction

When dividing by a fraction, we multiply by its reciprocal. So $\frac{\frac{7}{33}}{\frac{20}{33}}=\frac{7}{33}\times\frac{33}{20}=\frac{7}{20}$.

Answer:

$\frac{7}{20}$