An urn contains a total of 9 marbles out of which 2 are red, 3 are white, and the remaining are blue. Two marbles are drawn out of the urn in succession. What is the probability that both marbles are the same color if:
a. Replacement is allowed?
b.The first marble is not replaced?
To calculate the probability in both scenarios, we need to consider the number of favorable outcomes and the total number of possible outcomes.
a. Replacement is allowed:
When replacement is allowed, after drawing each marble out, we put it back into the urn before drawing the next marble. This means that the number of marbles in the urn remains the same for each draw. So, the probability of drawing a marble of the same color is calculated independently for each draw.
Since there are two red marbles, three white marbles, and four blue marbles in the urn, the total number of possible outcomes is 9.
To calculate the probability of drawing two marbles of the same color with replacement allowed, we need to find the probability of drawing two red, two white, or two blue marbles.
1. Probability of drawing two red marbles:
The probability of drawing the first red marble is 2/9. After replacing the first marble, the probability of drawing the second red marble is also 2/9. Thus, the probability of drawing two red marbles is (2/9) * (2/9) = 4/81.
2. Probability of drawing two white marbles:
Similarly, the probability of drawing the first white marble is 3/9 (or simplified as 1/3). After replacing the first marble, the probability of drawing the second white marble is also 3/9 (or 1/3). Thus, the probability of drawing two white marbles is (1/3) * (1/3) = 1/9.
3. Probability of drawing two blue marbles:
The probability of drawing the first blue marble is 4/9. After replacing the first marble, the probability of drawing the second blue marble is also 4/9. Thus, the probability of drawing two blue marbles is (4/9) * (4/9) = 16/81.
Adding up the probabilities of all possible outcomes, we get (4/81) + (1/9) + (16/81) = 45/81, which can be simplified as 5/9. So, the probability of drawing two marbles of the same color with replacement allowed is 5/9.
b. The first marble is not replaced:
When the first marble is not replaced, the number of marbles in the urn decreases after each draw, which affects the probability calculations.
1. Probability of drawing two red marbles:
The probability of drawing the first red marble is 2/9. After the first draw, there is one less marble in the urn, so the probability of drawing a second red marble is 1/8. Thus, the probability of drawing two red marbles is (2/9) * (1/8) = 1/36.
2. Probability of drawing two white marbles:
The probability of drawing the first white marble is 3/9 (or simplified as 1/3). After the first draw, there is one less white marble in the urn, so the probability of drawing a second white marble is 2/8 (or simplified as 1/4). Thus, the probability of drawing two white marbles is (1/3) * (1/4) = 1/12.
3. Probability of drawing two blue marbles:
The probability of drawing the first blue marble is 4/9. After the first draw, there are three blue marbles in the urn, so the probability of drawing a second blue marble is 3/8. Thus, the probability of drawing two blue marbles is (4/9) * (3/8) = 1/6.
Adding up the probabilities of all possible outcomes, we get (1/36) + (1/12) + (1/6) = 1/6. So, the probability of drawing two marbles of the same color with the first marble not replaced is 1/6.
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