I was reading a tutorial on marginal densities when I came across this example (rephrased).
A person is crossing the street and we want to compute the probability when he gets hit by a passing car depending on the color of the traffic light.
Let H be whether the person gets hit or not, and L be the color of the traffic light.
So $H = \{\text{hit, not hit} \}$ and $L = \{\text{red, yellow, green} \}$.
The probability of getting hit given that the light is red can be written as: $P(H = \text{hit}| L = \text{red})$. Clearly this is a conditional probability.
The probability of getting hit regardless of whatever the light is can be written as: $P(H = \text{hit})$. This is marginal, as I recently understood.
How can you say: $P(H,L)$. This is a joint probability. How do you translate it to a 'layman's sentence? How is it different from "The probability of getting hit AND the light is red"?
Thanks for your insights.
Best Answer
You actually had your answer right there.
$P(H=hit)$ is the marginal probability. It reads "The probability of getting hit.". It is the proportion of people that got hit crossing the street, irrespective of traffic light.
$P(H=hit|L=red)$ is the conditional probability. It reads "The probability that you get hit, given that the light is red". It is the proportion of hits among the people that cross the street in red light.
Finally, $P(H=hit, L=red)$ is the joint probability. It reads "the probability that a person gets hit by a car and that the light is red". It is the proportion of hits in red light among all people.
You certainly know the relationship
$P(H=hit, L=red) = P(H=hit | L=red) * P(L=red)$
In "layman's parlance", we can look at it as follows. Assume that the probability of having a red light is extremely small, but that people always get hit when crossing in red light. Let us assume you are an observer at the side of the street. You will see people getting hit, and rarely will you see the light turning red. Out of all people that cross the street, the chance they will get hit in red light is very tiny, since they almost never have that opportunity ($P(H=hit,L=red)$ is small because a red light is rare). However, if you observe long enough, you will eventually see people getting hit in red light, and notice that whenever the light is red, people crossing the street will get hit for sure ($P(H=hit|L=red)=1$).