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Q 29: Suppose that the probability that a pedestrian will be hit by a car while crossing the road at a pedestrian crossing without paying attention to the traffic light is to be computed. Let H be a discrete random variable taking one value from {Hit, Not Hit}. Let L be a discrete random variable taking one value from {Red, Yellow, Green}.

Realistically, H will be dependent on L. That is, P(H = Hit) and P(H = Not Hit) will take different values depending on whether L is red, yellow or green. A person is, for example, far more likely to be hit by a car when trying to cross while the lights for cross traffic are green than if they are red. In other words, for any given possible pair of values for H and L, one must consider the joint probability distribution of H and L to find the probability of that pair of events occurring together if the pedestrian ignores the state of the light.

Here is a table showing the conditional probabilities of being hit, depending on the state of the lights. (Note that the columns in this table must add up to 1 because the probability of being hit or not hit is 1 regardless of the state of the light.)

Conditional distribution: P(H|L)

 

L=Green

L=Yellow

L=Red

H=Not Hit

0.99

0.9

0.2

H=Hit

0.01

0.1

0.8

To find the joint probability distribution, we need more data. Let's say that P(L=green) = 0.2, P(L=yellow) = 0.1, and P(L=red) = 0.7. Multiplying each column in the conditional distribution by the probability of that column occurring, we find the joint probability distribution of H and L, given in the central 2×3 block of entries. (Note that the cells in this 2×3 block add up to 1).

Joint distribution: P(H,L)

 

 

L=Green

L=Yellow

L=Red

Marginal probability P(H)

H=Not Hit

0.198

0.09

0.14

0.428

H=Hit

0.002

0.01

0.56

0.572

Total

0.2

0.1

0.7

1

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