Final answer:
To calculate the probability that event A occurs exactly 42 times out of 105 independent observations, we can use the binomial probability formula.
Step-by-step explanation:
To calculate the probability that event A occurs exactly 42 times out of 105 independent observations, we can use the binomial probability formula.
The binomial probability formula is given by P(X = k) = (n choose k) * (p^k) * (q^(n-k)), where:
- P(X = k) is the probability of getting exactly k successes
- n is the total number of observations
- p is the probability of success for each observation
- q is the probability of failure for each observation, which is equal to 1 - p
- (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!)
Plugging in the values from the question, we have:
P(X = 42) = (105 choose 42) * (0.4^42) * (0.6^(105-42))
Using a calculator or software, we can calculate this probability.