Mathematics, 04.03.2021 19:30 steven2996

The probability of dry weather (D) = 1⁄2, rain (R) = 1⁄3 and snow (S) = 1⁄6 If it is dry the probability that someone will arrive on time is 4
/5.
If it rains the probability that someone will arrive on time is 2
/5.
If it snows the probability that someone will arrive on time is 1
/10.

Calculate the probability that somebody is late.

Answers: 2

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Answer from: ianmorales4894

9/20

Explanation:

Define the following events

D = weather is dryR = weather is rainyS = weather is snowyL = person is lateT = person is on time

We are given the following probabilities

P(D) = 1/2P(R) = 1/3P(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?"

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".

To recap so far, your teacher told you these three conditional probabilities

P(T given D) = 4/5P(T given R) = 2/5P(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) = 1/5P(L given R) = 3/5P(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.

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) = 1/10P(L and R) = P(L given R)*P(R) = (3/5)*(1/3) = 1/5P(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

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