Final answer:
The probability of 3 or more drivers out of 10 being involved in a car accident, given a 9% chance for each driver, is a binomial probability calculation. It involves summing the probabilities of having 3, 4, ..., up to 10 drivers involved in accidents. This calculation is vital for an insurance company to determine risk and premium costs.
Step-by-step explanation:
The question involves finding the probability of 3 or more drivers being involved in a car accident last year among 10 drivers randomly selected, given that 9% of all drivers were involved in a car accident. This is a binomial probability problem, where we can use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1 - p)^(n-k)
Where:
- n = total number of trials
- k = number of successful trials
- p = probability of success on a single trial
To solve for the probability of getting 3 or more drivers, we sum the probabilities of getting 3, 4, ..., up to 10 accidents:
P(X ≥ 3) = P(X = 3) + P(X = 4) + ... + P(X = 10)
The probability p of a driver being involved in an accident is 0.09 (or 9%), and the number of trials n is 10. We need to calculate the probability for each number of accidents from 3 to 10 and sum them to find the total probability.
It's worth noting that, for high values of n or extreme proportions, this calculation can become cumbersome, so using technology, such as a binomial calculator or statistical software, can be helpful.
Understanding probabilities is important for insurance companies to set premiums and for customers to understand their coverage and risks. The calculated probabilities can help the insurance company determine how to distribute risks and decide on the cost of premiums for the insured drivers.