Final answer:
Using a credit card for air miles and charging more than $2,000 per month are not independent events, as the combined probability does not equal the product of their individual probabilities.
Step-by-step explanation:
To determine whether using a credit card that gives a mile of air travel for each dollar spent and charging more than $2,000 per month are independent events, we need to analyze the provided probabilities.
Two events are independent if the occurrence of one does not affect the probability of the occurrence of the other. In mathematical terms, events A and B are independent if and only if P(A and B) = P(A) × P(B).
According to the information given:
- The probability of using a credit card that gives a mile per dollar is 35% (P(A)).
- The probability of charging more than $2,000 is 30% (P(B)).
- The probability of both using this type of credit card and charging more than $2,000 is 80% of the 30% who charge more than $2,000, which is 24% (P(A and B)).
Calculating the theoretical probability of P(A and B) if A and B were independent gives us:
P(A) × P(B) = 0.35 × 0.30 = 0.105 or 10.5%
However, the observed probability of P(A and B) given in the problem is 24%. Since 24% is not equal to 10.5%, the events are not independent.