Question

Curious about people's recycling behaviors, Emily put on some gloves and sifted through some recycling and trash bins. She kept count of the plastic type of each bottle and which bottles are properly dispensed Correctly placed Incorrectly placed Plastic #2 5 5 5 1 Plastic #4 What is the probability that a randomly selected bottle is correctly placed given that the bottle is made of plastic #4? Simplify any fractions.

Answer 1

`\begin{gathered} P(A|B)=\text{ Probability that randomly selected bottle }i\text{s correct placed given } \\ that\text{ }the\text{ bottle is made }of\text{ plastic \#4} \\ P(A)=\text{ Probability that randomly selected bottle }i\text{s correct placed} \\ P(B)=\text{ Probability }the\text{ bottle is made }of\text{ plastic \#4} \\ P(A\cap B)=\text{Probability that randomly selected bottle }i\text{s correct placed and } \\ \text{ }the\text{ bottle is made }of\text{ plastic \#4} \\ \\ P(B)=\text{ }(6)/(16)=(3)/(8) \\ \\ P(A\cap B)=(5)/(16) \\ \\ P(A|B)=(P(A\cap B))/(P(B)) \\ \\ P(A|B)=(5)/(16)\text{ }(\cdot)/(\cdot)\text{ }(3)/(8) \\ \\ P(A|B)=(5)/(6) \\ \\ \text{ Probability that randomly selected bottle }i\text{s correct placed given } \\ that\text{ }the\text{ bottle is made }of\text{ plastic \#4 is }(5)/(6) \end{gathered}`