Final answer:
Events C and D are independent. The probability of the union of events C and D (P(C∪D)) is .825.
Step-by-step explanation:
Let event C = taking an English class. Let event D = taking a speech class. Suppose P(C) = .75, P(D) = .3, P(C|D) = .75 and P(C AND D) = .225.
To determine if events C and D are independent, we need to check if the probability of C given D is equal to the probability of C. In this case, P(C|D) = .75, while P(C) = .75. Since these two probabilities are equal, we can conclude that events C and D are independent.
Now, to find the probability of the union of events C and D (P(C∪D)), we can use the formula: P(C∪D) = P(C) + P(D) - P(C AND D).
Substituting the given values, we have: P(C∪D) = .75 + .3 - .225 = .825.
the probability of the union of events C and D, represented as P(C∪D). In probability theory, the probability of the union of two events can be found using the inclusion-exclusion principle. The formula to calculate the probability of the union of two events C and D is as follows: \[ P(C ∪ D) = P(C) + P(D) - P(C ∩ D) \]
This formula states that the probability of the union of two events is the sum of the probabilities of each event minus the probability of their intersection (since the intersection is included in the probabilities of each individual event, we must subtract it to avoid counting it twice).
Given the probabilities: \[ P(C) = 0.44 \] \[ P(D) = 0.61 \] \[ P(C ∩ D) = 0.23 \] Let's plug these values into the formula: \[ P(C ∪ D) = 0.44 + 0.61 - 0.23 \] Now, we can calculate the value: \[ P(C ∪ D) = 1.05 - 0.23 \] \[ P(C ∪ D) = 0.82 \] So, the probability of the union of events C and D, P(C∪D), is 0.82.