Q. Small cars get better gas mileage, but they are not as safe as bigger cars. Small cars accounted for 18% of the vehicles on the road, but accidents involving small cars led to 11,898 fatalities during a recent year. Assume the probability a small car is involved in an accident is .18. The probability of an accident involving a small car leading to a fatality is .128 and the probability of an accident not involving a small car leading to a fatality is .05. Suppose you learn of an accident involving a fatality. What is the probability a small car was involved? Assume that the likelihood of getting into an accident is independent of car size.
Answers
Answer:
Question
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Small cars get better gas mileage, but they are not as safe as bigger cars. Small cars accounted
for 18% of the vehicles on the road, but accidents involving small cars led to 11,898 fatalities
during a recent year (Reader’s Digest, May 2000). Assume the probability a small car is
involved in an accident is .18. The probability of an accident involving a small car leading to
a fatality is .128 and the probability of an accident not involving a small car leading to a fatality
is .05. Suppose you learn of an accident involving a fatality. What is the probability a
small car was involved? Assume that the likelihood of getting into an accident is independent
of car size.
Explanation
Verified
Step 1
1 of 3
The task states following probabilities in case of traffic accidents :
\begin{gather*} P(\text{small car}) = 0.18\\ P(\text{accident}| \text{ small car}) =0.18\\ P(\text{fatality}| \text{ small car}) =0.128\\ P(\text{fatality}| \text{ not small car}) =0.05\\ \end{gather*}
P(small car)=0.18
P(accident∣ small car)=0.18
P(fatality∣ small car)=0.128
P(fatality∣ not small car)=0.05
Use the complement rule to define the probability of not having a small car
\begin{gather*} P(not A) = 1 - P(A) \end{gather*}
P(notA)=1−P(A)
Put the known parameters in the rule:
\begin{gather*} P(\text{not small car}) = 1 - P(\text{small car}) =1-0.18=0.82 \end{gather*}
P(not small car)=1−P(small car)=1−0.18=0.82
Since the events of accidents are independent of the car size, rule sets following:
\begin{gather*} P(\text{accident}| \text{ small car}) = P(\text{accident}| \text{ not small car})\\ P(\text{accident}| \text{ not small car}) = 0.18 \end{gather*}
P(accident∣ small car)=P(accident∣ not small car)
P(accident∣ not small car)=0.18
Bayes' theorem says:
\begin{gather*} P(A_i | B) =\frac{P(A_i)P(B|Ai)}{P(A_1)P(B|A1) + ... + P(A_k)P(B|Ak)} \end{gather*}
P(A
i
∣B)=
P(A
1
)P(B∣A1)+...+P(A
k
)P(B∣Ak)
P(A
i
)P(B∣Ai)
First compute the probability of an accident involving a fatality:
\begin{gather*} P(S)=P(\text{accident}| \text{ small car}) \cdot P(\text{fatality}| \text{ small car}) = 0.18 \cdot 0.128 = 0.02304\\ P(B)=P(\text{accident}| \text{ not small car}) \cdot P(\text{fatality}| \text{ not small car}) = 0.18 \cdot 0.05 = 0.009\\ \end{gather*}
P(S)=P(accident∣ small car)⋅P(fatality∣ small car)=0.18⋅0.128=0.02304
P(B)=P(accident∣ not small car)⋅P(fatality∣ not small car)=0.18⋅0.05=0.009
Step 2
2 of 3
Follow the Bayes rule and compute:
\begin{gather*} P(\text{small car }| \text{fatality}) = \frac{P(\text{small car})\cdot P(S)}{P(\text{small car})\cdot P(S) + P(\text{not small car})\cdot P(B)}\\ P(\text{small car }| \text{fatality}) = \frac{0.18\cdot 0.02304}{0.18\cdot 0.02304+0.82\cdot 0.009}\\ P(\text{small car }| \text{fatality}) = \frac{0.0041}{0.0041+0.0074}=0.3571 \end{gather*}
P(small car ∣fatality)=
P(small car)⋅P(S)+P(not small car)⋅P(B)
P(small car)⋅P(S)
P(small car ∣fatality)=
0.18⋅0.02304+0.82⋅0.009
0.18⋅0.02304
P(small car ∣fatality)=
0.0041+0.0074
0.0041
=0.3571
Result
3 of 3
The probability that a small car is involved is 0.3571.
Explanation:
Answer:
The probability (small car involvement) is 0.3571.
Explanation
Given
Solution
Probability small car=0.18
probability of accidental small car=.18
Probability of not small car=0.05
According to the complement rule, P(NA)=1-P(A)
Probability of not small car=1-Probability of the small car
=1-0.18=0.82
As per the question accidents are not based on the size of the car.
P(accident small car)=0.18
P(A/B)=P(B/A)P(A)/P(B)
Probability of involving a fatal accident P(S)=P(accidental small
car)×P(fatal small car)
=0.18×0.128=0.02304
=0.18×0.05=0.009
p(B)=P(accidental not small
P(small car fatality)=P(small car)×P(S)+P(not small car)×P(B)P(small
=car)×P(S)
=0.18×0.02304+0.82×.009
=0.0041
Probability of small car fatality=0.3571
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