a certain bag of marbles has marbles of different materials (silver, gold, copper) and sizes (small, medium…

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})