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.
a. What is the probability of selecting a purple marble and then a white marble?
b. What is the probability of selecting two white marbles?
c. Is there a greater chance of selecting two white marbles in a row or two purple marbles in a row? Show your work. Make sure you answer each question in as few steps as possible
a. The probability of selecting a purple marble is 7/9, since there are 7 purple marbles out of 9 total. After selecting a purple marble, there are now 1 white marble and 7 purple marbles left in the bag. The probability of selecting a white marble from this reduced set is 1/8. Therefore, the probability of selecting a purple marble and then a white marble is (7/9) x (1/8) = 7/72.
b. The probability of selecting a white marble on the first draw is 2/9. After selecting a white marble and not replacing it, there is now 1 white marble and 7 purple marbles left in the bag. The probability of selecting a white marble from this reduced set is 1/8. Therefore, the probability of selecting two white marbles is (2/9) x (1/8) = 1/36.
c. The probability of selecting two white marbles in a row is 1/36, as calculated in part (b). The probability of selecting two purple marbles in a row can be calculated as follows: the probability of selecting a purple marble on the first draw is 7/9. After selecting a purple marble and not replacing it, there are now 6 purple marbles left in the bag out of a total of 8 remaining marbles. Therefore, the probability of selecting another purple marble is 6/8 = 3/4. The probability of selecting two purple marbles in a row is (7/9) x (3/4) = 21/36. Since 21/36 is greater than 1/36, there is a greater chance of selecting two purple marbles in a row than two white marbles in a row.
a. The probability of selecting a purple marble and then a white marble can be calculated by multiplying the probability of selecting a purple marble and the probability of selecting a white marble from the remaining marbles.
Step 1: Calculate the probability of selecting a purple marble: There are 7 purple marbles out of 9 total marbles, so the probability is 7/9.
Step 2: Calculate the probability of selecting a white marble from the remaining marbles: After selecting a purple marble, there are now 8 marbles left, including 2 white marbles. So the probability is 2/8.
Step 3: Multiply the probabilities: (7/9) * (2/8) = 14/72 = 7/36
So, the probability of selecting a purple marble and then a white marble is 7/36.
b. The probability of selecting two white marbles can be calculated by multiplying the probability of selecting a white marble and then, without replacement, the probability of selecting another white marble from the remaining marbles.
Step 1: Calculate the probability of selecting a white marble: There are 2 white marbles out of 9 total marbles, so the probability is 2/9.
Step 2: Calculate the probability of selecting another white marble from the remaining marbles: After selecting a white marble, there is now 1 white marble left out of 8 remaining marbles. So the probability is 1/8.
Step 3: Multiply the probabilities: (2/9) * (1/8) = 2/72 = 1/36
So, the probability of selecting two white marbles is 1/36.
c. To determine whether there is a greater chance of selecting two white marbles in a row or two purple marbles in a row, we compare the probabilities calculated in parts a and b.
The probability of selecting two white marbles in a row is 1/36, and the probability of selecting two purple marbles in a row can be calculated in a similar way as part a:
Step 1: Calculate the probability of selecting a purple marble: There are 7 purple marbles out of 9 total marbles, so the probability is 7/9.
Step 2: Calculate the probability of selecting another purple marble from the remaining marbles: After selecting a purple marble, there are now 6 purple marbles left out of 8 remaining marbles. So the probability is 6/8.
Step 3: Multiply the probabilities: (7/9) * (6/8) = 42/72 = 7/12
So, the probability of selecting two purple marbles in a row is 7/12.
Comparing 1/36 and 7/12, we can see that the probability of selecting two purple marbles in a row (7/12) 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.