Questions: Fatal Accidents The American Automobile Association (AAA) reports that of the fatal car and truck accidents, 25% occurred while the driver was using a cell phone. If 3 accidents are chosen at random, find the probability for the following. Please round the final answers to 2 or 3 decimal places. Part 1 of 3 (a) All occurred while the driver was using a cell phone. P (all occurred while the driver was using a cell phone) = 0.016 Part 2 of 3 (b) None occurred while the driver was using a cell phone. P (none occurred while the driver was using a cell phone) =

Solution Steps

We are tasked with finding the probability that none of the 3 randomly chosen fatal accidents occurred while the driver was using a cell phone. Given that the probability of a driver using a cell phone during an accident is \( p = 0.25 \), the probability of not using a cell phone is \( q = 1 - p = 0.75 \).

The probability of exactly \( x \) successes (in this case, none of the accidents involving cell phone use) in \( n \) trials can be calculated using the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

For our scenario:

• \( n = 3 \) (the number of accidents)
• \( x = 0 \) (the number of accidents where the driver was using a cell phone)

Substituting the values into the formula:

\[ P(X = 0) = \binom{3}{0} \cdot (0.25)^0 \cdot (0.75)^{3} \]

Calculating each component:

• \( \binom{3}{0} = 1 \)
• \( (0.25)^0 = 1 \)
• \( (0.75)^{3} = 0.421875 \)

Thus, we have:

\[ P(X = 0) = 1 \cdot 1 \cdot 0.421875 = 0.421875 \]

Rounding this to three decimal places gives us:

\[ P(X = 0) \approx 0.422 \]

Final Answer