You roll a red number cube and a blue number cube. Let A be the event "at least one number cube shows a 6"
a) Find P(A) by finding the sum of P(6 on red cube, no 6 on blue cube), P(no 6 on red cube, 6 on blue cube), and P(6 on both cubes)
b) Describe the event "not A" then find P(not A) and use it to find P(A)
c) Compare the methods in part a and b.
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a) To find P(A), we need to calculate the probabilities of the three events mentioned:
1. P(6 on red cube, no 6 on blue cube): Since the red cube has 6 faces, the probability of rolling a 6 on the red cube is 1/6. The probability of not rolling a 6 on the blue cube is 5/6. Therefore, the probability of this event is (1/6) * (5/6) = 5/36.
2. P(no 6 on red cube, 6 on blue cube): The probability of not rolling a 6 on the red cube is 5/6, and the probability of rolling a 6 on the blue cube is 1/6. So, the probability of this event is (5/6) * (1/6) = 5/36.
3. P(6 on both cubes): The probability of rolling a 6 on both the red and blue cubes is (1/6) * (1/6) = 1/36.
To find P(A), we sum up the probabilities of the three events: P(A) = (5/36) + (5/36) + (1/36) = 11/36.
b) The event "not A" refers to the situation where neither cube shows a 6. To find P(not A), we need to calculate the probability of not rolling a 6 on both cubes. The probability of not rolling a 6 on the red cube is 5/6, and the probability of not rolling a 6 on the blue cube is also 5/6. So, the probability of not A is (5/6) * (5/6) = 25/36.
To find P(A), we use the complement rule: P(A) = 1 - P(not A). So, P(A) = 1 - (25/36) = 11/36.
c) Both methods in part a and b yield the same answer for P(A), which is 11/36. The first method explicitly considers the three different possibilities that contribute to the event A, while the second method calculates the probability of the complement event, "not A," and then uses it to find P(A) using the complement rule. Both approaches can be used to find the probability and provide the same result.