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A bag contains 2 white marbles and 7 purple marbles. Two marbles are drawn at random. One marble is drawn and not replaced. Then a second marble is drawn.
What is the probability of selecting a purple marble and then a white marble?
What is the probability of selecting two white marbles?
Is there a greater chance of selecting two white marbles in a row or two purple marbles in a row? Show your work.
Probability of selecting a purple marble and then a white marble:
The probability of drawing a purple marble on the first draw is 7/9, since there are 7 purple marbles out of 9 total marbles in the bag. After drawing a purple marble and not replacing it, there will be 8 marbles left in the bag with 2 white ones. Therefore, the probability of drawing a white marble on the second draw is 2/8 or 1/4.
To find the probability of both events happening together, we multiply the probabilities:
(7/9) x (1/4) = 7/36
Therefore, the probability of selecting a purple marble and then a white marble is 7/36.
Probability of selecting two white marbles:
The probability of selecting a white marble on the first draw is 2/9, since there are 2 white marbles out of 9 total marbles in the bag. After drawing a white marble and not replacing it, there will be 8 marbles left in the bag with only 1 white one. Therefore, the probability of drawing another white marble on the second draw is 1/8.
To find the probability of both events happening together, we multiply the probabilities:
(2/9) x (1/8) = 1/36
Therefore, the probability of selecting two white marbles is 1/36.
Is there a greater chance of selecting two white marbles in a row or two purple marbles in a row?
To compare the probabilities, we need to find the probability of selecting a purple marble and then another purple marble.
The probability of drawing a purple marble on the first draw is 7/9. After drawing a purple marble and not replacing it, there will be 8 marbles left in the bag with 7 purple ones. Therefore, the probability of drawing another purple marble on the second draw is 7/8.
To find the probability of both events happening together, we multiply the probabilities:
(7/9) x (7/8) = 49/72
Therefore, the probability of selecting two purple marbles in a row is 49/72, which is greater than the probability of selecting two white marbles in a row (1/36).
Therefore, there is a greater chance of selecting two purple marbles in a row than two white marbles in a row.
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1. Probability of purple then white = (7/9) x (1/4) = 7/36
2. Probability of two white marbles in a row = (2/9) x (1/8) = 1/36
3. Probability of two purple marbles in a row = (7/9) x (7/8) = 49/72
Therefore, the probability of selecting two purple marbles in a row is greater than the probability of selecting two white marbles in a row.
To find the probability, we need to determine the number of favorable outcomes and the total number of possible outcomes.
Let's start with the first question:
What is the probability of selecting a purple marble and then a white marble?
Step 1: Determine the number of favorable outcomes.
For the first draw, there are 7 purple marbles out of a total of 9 marbles. Therefore, the number of favorable outcomes for the first draw is 7.
For the second draw, after a purple marble has been drawn and not replaced, there are now 8 marbles left in the bag. Out of these, 2 are white marbles. Therefore, the number of favorable outcomes for the second draw is 2.
Step 2: Determine the total number of possible outcomes.
For the first draw, there are a total of 9 marbles in the bag.
For the second draw, after a marble has been drawn and not replaced, there are now 8 marbles left in the bag.
Step 3: Calculate the probability.
To find the probability of two events happening in succession, we multiply the probabilities of each event.
The probability of selecting a purple marble on the first draw is 7/9.
The probability of selecting a white marble on the second draw, given that a purple marble was drawn on the first draw, is 2/8 (since there are 2 white marbles out of 8 remaining marbles).
Therefore, the probability of selecting a purple marble and then a white marble is (7/9) * (2/8) = 7/36.
Now let's move to the second question:
What is the probability of selecting two white marbles?
Step 1: Determine the number of favorable outcomes.
For the first draw, there are 2 white marbles out of a total of 9 marbles. Therefore, the number of favorable outcomes for the first draw is 2.
For the second draw (without replacement), there is 1 white marble left in the bag. Therefore, the number of favorable outcomes for the second draw is 1.
Step 2: Determine the total number of possible outcomes.
For the first draw, there are a total of 9 marbles in the bag.
For the second draw, after a marble has been drawn and not replaced, there are now 8 marbles left in the bag.
Step 3: Calculate the probability.
To find the probability of two white marbles being selected, we multiply the probabilities of each event.
The probability of selecting a white marble on the first draw is 2/9.
The probability of selecting a white marble on the second draw, given that a white marble was drawn on the first draw, is 1/8 (since there is 1 white marble out of 8 remaining marbles).
Therefore, the probability of selecting two white marbles is (2/9) * (1/8) = 1/36.
Now let's move to the third question:
Is there a greater chance of selecting two white marbles in a row or two purple marbles in a row?
To compare the probabilities, we need to calculate the probability of selecting two purple marbles in a row.
The probability of selecting two purple marbles is (7/9) * (6/8) = 7/12.
Comparing the fractions, we can see that the probability of selecting two white marbles (1/36) is smaller than the probability of selecting two purple marbles (7/12).
Therefore, there is a greater chance of selecting two purple marbles in a row than two white marbles in a row.