Question

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{3}{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: Calculate \( P(\text{blue})\times P(\text{green}) \)

Given \( P(\text{blue})=\frac{3}{10} \) and \( P(\text{green})=\frac{1}{5} \).

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Convert \( \frac{3}{50} \) to a fraction with denominator 100: \( \frac{3}{50}=\frac{3\times2}{50\times2}=\frac{6}{100} \)

Step2: Compare \( P(\text{blue})\times P(\text{green}) \) and \( P(\text{blue and green}) \)

We have \( P(\text{blue})\times P(\text{green})=\frac{6}{100} \) and \( P(\text{blue and green})=\frac{3}{100} \)
Since \( \frac{6}{100}
eq\frac{3}{100} \), i.e., \( P(\text{blue})\times P(\text{green})
eq P(\text{blue and green}) \)

Answer:

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