Final answer:
In an experiment, two events A and B are given. We need to find the probability of event B, the probability of A happening but not B, and the probability of A or not B happening. We also need to determine whether events A and B are independent. Using the given information, we can calculate the probabilities and conclude that A and B are not independent.
Step-by-step explanation:
(a) To find P[B], we can use the formula P[A U B] = P[A] + P[B] - P[A n B], where P[A U B] is the probability of A or B happening, P[A] is the probability of A happening, P[B] is the probability of B happening, and P[A n B] is the probability of A and B happening at the same time. Given that P[A U B] = 5/8 and P[A] = 3/8, we can substitute these values into the formula to get 5/8 = 3/8 + P[B] - P[A n B]. Since A and B are mutually exclusive, meaning they cannot occur together, P[A n B] = 0. Therefore, 5/8 = 3/8 + P[B]. Solving for P[B], we get P[B] = 2/8 = 1/4.
To find P[A n B'], we can use the formula P[A n B'] = P[A] - P[A n B], where P[A n B'] is the probability of A happening but not B, P[A] is the probability of A happening, and P[A n B] is the probability of A and B happening at the same time. Given that P[A] = 3/8 and P[A n B] = 0 (since A and B are mutually exclusive), we can substitute these values into the formula to get P[A n B'] = 3/8 - 0 = 3/8.
To find P[A U B'], we can use the formula P[A U B'] = P[A] + P[B'], where P[A U B'] is the probability of A or B' happening, P[A] is the probability of A happening, and P[B'] is the probability of B' happening (which is equal to 1 - P[B]). Given that P[A] = 3/8 and P[B] = 1/4, we can substitute these values into the formula to get P[A U B'] = 3/8 + (1 - 1/4) = 3/8 + 3/4 = 9/8. However, probabilities cannot exceed 1, so we can conclude that P[A U B'] = 1.
(b) A and B are not independent events. Two events are independent if the occurrence of one event does not affect the probability of the other event. In this case, if A and B were independent, P[B] would be equal to P[B|A], which is the probability of B happening given that A has already happened. However, P[B] = 1/4 and P[B|A] = P[A n B] / P[A] = 0 / (3/8) = 0, which are not equal. Therefore, we can conclude that A and B are not independent.