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Chapter 9: Problem 34

Find the probability for the experiment of drawing two marbles at random (without replacement) from a bag containing one green, two yellow, and three red marbles. The marbles are different colors.

Short Answer

Expert verified
The probabilities for the different combinations of drawing two marbles from the bag are: \nP(Red, Red) = 3/15 or 0.20 \nP(Yellow, Yellow) = 1/15 or 0.067 \nP(Green, Other color) = 5/15 or 0.333.

Step by step solution

01

Identify Total Outcomes

First, the total number of outcomes for drawing two marbles from the bag should be identified. There are total 6 marbles in the bag, so there are \( \binom{6}{2} \) which is 15 total outcomes(call the total outcomes as 'T').
02

Analyze the Desired Events

Next, analyze the different combinations of drawing 2 marbles from the bag. Total there are 3 combination events - either draw 2 red, or 2 yellow, or 1 green and 1 other color. Let's calculate the total possible outcomes for each event: \n- Draw 2 reds, there are \( \binom{3}{2} \) = 3 outcomes (call this event `R`). \n- Draw 2 yellows there is \( \binom{2}{2} \) = 1 outcome (call this event `Y`). \n- Draw 1 green and 1 other color marble, there are \( \binom{1}{1} \) * \( \binom{5}{1} \) = 5 outcomes (call this event `G`).
03

Calculate Probability

Finally, to find the probability of each combination, divide the number of outcomes in each event by the total number of outcomes. So the probabilities are: \nP(R) = 3/15, \nP(Y) = 1/15, \nP(G) = 5/15.

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