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Answer: 9/20</h3>
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Explanation:
Define the following events
- D = weather is dry
- R = weather is rainy
- S = weather is snowy
- L = person is late
- T = person is on time
We are given the following probabilities
- P(D) = 1/2
- P(R) = 1/3
- P(S) = 1/6
Then we're further told that "if it is dry, then the probability someone will arrive on time is 4/5". We can condense that into the notation
P(T given D) = 4/5
The "given D" means that we know 100% that event D has occurred. So we're seeing how event T behaves because of D occurring. In other words P(T given D) is the same as saying "What is P(T) when we know D has happened?"
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The instructions also mention that "If it rains, the probability someone is on time is 2/5" tells us we have the notation
P(T given R) = 2/5
And also, the probability someone is on time if it is snowy is
P(T given S) = 1/10
due to it saying "If it snows, the probability that someone will arrive on time is 1/10".
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To recap so far, your teacher told you these three conditional probabilities
- P(T given D) = 4/5
- P(T given R) = 2/5
- P(T given S) = 1/10
They are conditional due to the "given" as part of the probability. They depend on the given event happening.
Subtract each fraction from 1 and we end up with these complementary probabilities
- P(L given D) = 1/5
- P(L given R) = 3/5
- P(L given S) = 9/10
I'm using the idea that P(A)+P(B) = 1, where A and B are complementary events. One or the other event must happen. Either you are on time, or you are late.
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Recall the conditional probability formula is
P(A given B) = P(A and B)/P(B)
which rearranges to
P(A and B) = P(A given B)*P(B)
We'll use this idea to get the following
- P(L and D) = P(L given D)*P(D) = (1/5)*(1/2) = 1/10
- P(L and R) = P(L given R)*P(R) = (3/5)*(1/3) = 1/5
- P(L and S) = P(L given S)*P(S) = (9/10)*(1/6) = 3/20
This means,
P(L) = P(L and D) + P(L and R) + P(L and S) ..... law of total probability
P(L) = 1/10 + 1/5 + 3/20
P(L) = 2/20 + 4/20 + 3/20
P(L) = (2+4+3)/20
P(L) = 9/20 is the probability someone is late
