Question

a certain bag of marbles has marbles of different materials (silver, gold, copper) and sizes (small, medium, large). consider the following probabilities for drawing a random marble from the jar: p(silver) = 2/5 p(small) = 1/4 p(silver u small) = 11/20 p(silver ∩ small) = 1/10 p(silver | small) = 2/5 based on the information given, are the events silver and small independent? choose your answer. based on your previous answer, what is p(small | silver)? type your answer.

Explanation:

Step1: Recall the independence - condition

Two events \(A\) and \(B\) are independent if \(P(A\cap B)=P(A)\times P(B)\). Let \(A\) be the event of drawing a silver marble (\(P(A) = P(\text{silver})=\frac{2}{5}\)), and \(B\) be the event of drawing a small marble (\(P(B)=P(\text{small})=\frac{1}{4}\)). Then \(P(A)\times P(B)=\frac{2}{5}\times\frac{1}{4}=\frac{2}{20}=\frac{1}{10}\), and \(P(A\cap B)=P(\text{silver}\cap\text{small}) = \frac{1}{10}\).

Step2: Determine independence

Since \(P(\text{silver}\cap\text{small})=P(\text{silver})\times P(\text{small})=\frac{1}{10}\), the events Silver and Small are independent.

Step3: Use the formula for conditional - probability for independent events

For independent events \(A\) and \(B\), \(P(B|A)=\frac{P(A\cap B)}{P(A)}\), and since \(P(A\cap B) = P(A)\times P(B)\), then \(P(B|A)=P(B)\). So \(P(\text{small}|\text{silver})=P(\text{small})\).

Answer:

The events Silver and Small are independent.
\(P(\text{small}|\text{silver})=\frac{1}{4}\)