Final answer:
To find the probability that there will be at least one fatality for the next billion miles traveled, we can use the Poisson distribution. The rate of car fatalities is given as 33,561 fatalities for 2969 billion miles traveled. Using the Poisson distribution formula, we can calculate the probability of at least one fatality to be approximately 0.99999 (rounded to 5 decimal places).
Step-by-step explanation:
To find the probability that there will be at least one fatality for the next billion miles traveled, we can use the Poisson distribution. The Poisson distribution is used to model the number of events that occur in a fixed interval of time or space.
In this case, the rate of car fatalities is given as 33,561 fatalities for 2969 billion miles traveled. We can calculate the average rate of fatalities per billion miles by dividing the total fatalities by the total miles traveled: average rate = 33561/2969 = 11.3 fatalities per billion miles.
Now, we can use the Poisson distribution formula to find the probability of at least one fatality:
P(X >= 1) = 1 - P(X = 0) = 1 - e^(-λ) = 1 - e^(-11.3) ≈ 0.99999 (rounded to 5 decimal places)