Agent Smith said:
I imagined this scenario (it squares with the news article mentioned):
P(accident involving cars) = 0.8
P(dying in a car accident) = 0.3
P(accident involving trucks) = 0.2
P(dying in a truck accident) = 0.7
P(accident involving a car and dying in that accident) = P(C) = ##0.8 \times 0.3 = 0.24##
P(accident involving a truck and dying that accident) = P(T) = ## 0.2 \times 0.7 = 0.14##
Let's write things more clearly. Let ##C## be the event that there was an accident with a car, let ##T## be the event that there was an accident with a truck, and let ##D## be the event that the cyclist died. So we have $$P(C)=0.8$$$$P(D|C)=0.3$$$$P(T)=0.2$$$$P(D|T)=0.7$$ So by the definition of conditional probability $$P(D \cap C)=P(C) P(D|C)=0.24$$$$P(D\cap T)=P(T) P(D|T)=0.14$$
So, we can see that ##P(D|T)>P(D|C)## meaning that it is more likely that a cyclist dies given that they were in an accident with a truck than it is that a cyclist dies given that they were in a car accident. We can also see that ##P(D\cap C)>P(D\cap T)## meaning that it is more likely that a cyclist is in a car accident and dies than it is that a cyclist is in a truck accident and dies.
Agent Smith said:
Does P(C) > P(T) weaken/nullify the justification for more regulations on trucks?
No. Probabilities don't tell you what you should or shouldn't do, and they don't justify or un-justify regulations. They simply tell you risks and uncertainties. What you decide to do or not to do can be perfectly justified based on other considerations. This usually depends on your goals, your tolerance for risk, and the cost-benefit of all the possible choices.
Here, it is unclear what the goal of the proposed regulations is. I would guess that the goal is to reduce ##P(D)##, the probability of a cyclist dying. This is important to clarify because none of the above computations give you ##P(D)##. It is possible that a regulation could reduce ##P(D\cap T)## without changing ##P(D)##. It is also possible that a regulation could reduce ##P(D)## without changing either ##P(D\cap T)## or ##P(D\cap C)##.
Here, it is also unclear what the tolerance for risk is. This is sometimes known as the value of a statistical life or the value of preventing a fatality. In typical liberal democracies that is typically in the range of 1 to 10 million dollars. In other words, 1000 people would typically be willing to each pay $1000 to $10000 to reduce their individual risk of death by p=0.001.
Here, it is also unclear what the cost-benefit of the proposed regulations are and who bears the cost. Maybe three different regulations exist, all of which lead to the same decrease in ##P(D)## and one would cost $100 per car driver and the other would cost $1000 per truck driver and the last would cost $1000 per bicyclist. The total cost to society would depend on the number of car drivers, truck drivers, and cyclists. And the willingness to pay may differ for the different groups. Car drivers may not be willing to pay $100 to reduce the risk for cyclists while cyclists may be willing to pay $1000 to reduce the risk to themselves.
Probabilities cannot make decisions for you. They can just quantify risk.
