Question

determining if two events are independent or dependent given the following probabilities for choosing 2 marbles from a bag of 10 marbles, determine if the events are dependent or independent. p(blue) = (\frac{1}{10}), p(green) = (\frac{1}{5}), and p(blue and green) = (\frac{3}{100}). dependent, because the product of p(blue) and p(green) does not equal p(blue and green) dependent, because the product of p(blue) and p(green) equals p(blue and green) independent, because the product of p(blue) and p(green) does not equal p(blue and green) independent, because the product of p(blue) and p(green) equals p(blue and green)

Explanation:

Step1: Recall the independence rule

For two events \( A \) and \( B \), they are independent if \( P(A \cap B)=P(A)\times P(B) \). Here, \( A \) is "blue" and \( B \) is "green".

Step2: Calculate the product of \( P(\text{blue}) \) and \( P(\text{green}) \)

Given \( P(\text{blue})=\frac{1}{10} \) and \( P(\text{green})=\frac{1}{5} \). The product is \( \frac{1}{10}\times\frac{1}{5}=\frac{1\times1}{10\times5}=\frac{1}{50}=\frac{2}{100} \).

Step3: Compare with \( P(\text{blue and green}) \)

We know \( P(\text{blue and green})=\frac{3}{100} \). Since \( \frac{2}{100}
eq\frac{3}{100} \), the product of \( P(\text{blue}) \) and \( P(\text{green}) \) does not equal \( P(\text{blue and green}) \). By the definition of independent events, if \( P(A \cap B)
eq P(A)\times P(B) \), the events are dependent.

Answer:

dependent, because the product of \( P(\text{blue}) \) and \( P(\text{green}) \) does not equal \( P(\text{blue and green}) \)