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.)
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Conditional distribution: P(H|L) |
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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).
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Joint distribution: P(H,L) |
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|
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 |
