Final answer:
No, events e and f are not independent.
Step-by-step explanation:
To determine whether events e and f are independent, we first need to find the probabilities of each event and the probability of their intersection.
e is the event that a randomly generated bit string of length three contains an odd number of 1's. The possible outcomes are 001, 010, 011, 100, 101, and 111. There are 4 outcomes that satisfy event e. So, P(e) = 4/8 = 1/2.
f is the event that the string starts with 1. The possible outcomes are 100, 101, and 111. There are 3 outcomes that satisfy event f. So, P(f) = 3/8
The intersection of events e and f is the outcome 111, which satisfies both events. So, P(e ∩ f) = 1/8.
If e and f were independent, then P(e ∩ f) should be equal to P(e)P(f). However, P(e)P(f) = (1/2)(3/8) = 3/16, which is not equal to 1/8. Therefore, events e and f are not independent.