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Problem 1

# Three balls are selected at random without replacement from an urn containing four green balls and six red balls. Let the random variable $$X$$ denote the number of green balls drawn. a. List the outcomes of the experiment. b. Find the value assigned to each outcome of the experiment by the random variable $$X$$. c. Find the event consisting of the outcomes to which a value of 3 has been assigned by $$X$$.

Expert verified
a. The possible outcomes are: GGG, GGR, GRG, GRR, RGG, RGR, RRG, RRR. b. The values assigned by the random variable $$X$$: - GGG: $$X = 3$$ - GGR, GRG, RGG: $$X = 2$$ - GRR, RGR, RRG: $$X = 1$$ - RRR: $$X = 0$$ c. The event with $$X = 3$$ is {GGG}.
See the step by step solution

## Step 1: List all possible outcomes

Since we are drawing three balls, each draw can result in either a green ball (G) or a red ball (R). We can list the possible outcomes as follows: 1. GGG 2. GGR 3. GRG 4. GRR 5. RGG 6. RGR 7. RRG 8. RRR

## Step 2: Assign values to outcomes by the random variable $$X$$

The random variable $$X$$ represents the number of green balls drawn. We can assign values to the outcomes listed in Step 1 based on this: 1. GGG: $$X = 3$$ 2. GGR: $$X = 2$$ 3. GRG: $$X = 2$$ 4. GRR: $$X = 1$$ 5. RGG: $$X = 2$$ 6. RGR: $$X = 1$$ 7. RRG: $$X = 1$$ 8. RRR: $$X = 0$$

## Step 3: Find the event with $$X = 3$$

Now, we need to find the event consisting of the outcomes to which a value of 3 has been assigned by $$X$$. From the values assigned in Step 2, we can see that there is only one outcome with $$X = 3$$: 1. GGG: $$X = 3$$ Thus, the event to which a value of 3 has been assigned by $$X$$ is {GGG}.

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