Before each class, I either drink a cup of coffee, a cup of tea, or a...

Before each class, I either drink a cup of coffee, a cup of tea, or a cup of water. The probability of coffee is 0.7, the probability of tea is 0.2, and the probability of water is 0.1. If I drink coffee, the probability that the lecture ends early is 0.3. If I drink tea, the probability that the lecture ends early is 0.2. If I drink water, the lecture never ends early.
1) What’s the probability that I drink tea and ﬁnish the lecture early?
2) What’s the probability that I ﬁnish the lecture early?
3) Given the lecture ﬁnishes early, what’s the probability I drank coffee?

Solutions

Expert Solution

The probability of coffee, P[ Coffee ] = 0.7

The probability of tea, P[ Tea ] = 0.2

The probability of water, P[ Water ] = 0.1

P[ I drink coffee, the lecture ends early ] = P[ lecture ends early | coffee ] = 0.3

P[ I drink tea, the lecture ends early ] = P[ lecture ends early | tea ] = 0.2

P[ I drink water, the lecture ends early ] = P[ lecture ends early | water ] = 0

1) What’s the probability that I drink tea and ﬁnish the lecture early?

P[ I drink tea and ﬁnish the lecture early ] = P[ lecture ends early | tea ]*P[ Tea ]

P[ I drink tea and ﬁnish the lecture early ] = 0.2*0.2

P[ I drink tea and ﬁnish the lecture early ] = 0.04

2) What’s the probability that I ﬁnish the lecture early?

P[ I ﬁnish the lecture early ] = P[ I drink coffee and ﬁnish the lecture early ] + P[ I drink tea and ﬁnish the lecture early ] + P[ I drink water and ﬁnish the lecture early ]

P[ I drink coffee and ﬁnish the lecture early ] = P[ lecture ends early | coffee ]*P[ Coffee ]

P[ I drink coffee and ﬁnish the lecture early ] = 0.7*0.3

P[ I drink coffee and ﬁnish the lecture early ] = 0.21

P[ I drink tea and ﬁnish the lecture early ] = P[ lecture ends early | tea ]*P[ Tea ]

P[ I drink tea and ﬁnish the lecture early ] = 0.2*0.2

P[ I drink tea and ﬁnish the lecture early ] = 0.04

P[ I drink water and ﬁnish the lecture early ] = P[ lecture ends early | water ]*P[ Water ]

P[ I drink water and ﬁnish the lecture early ] = 0.1*0

P[ I drink water and ﬁnish the lecture early ] = 0

P[ I ﬁnish the lecture early ] = 0.21 +0.04 + 0

P[ I ﬁnish the lecture early ] = 0.25

3) Given the lecture ﬁnishes early, what’s the probability I drank coffee?

P[ I drank coffee | the lecture ﬁnishes early ] = P[ I drink coffee and ﬁnish the lecture early ]/P[ I ﬁnish the lecture early ]

P[ I drank coffee | the lecture ﬁnishes early ] = 0.21/0. 25

P[ I drank coffee | the lecture ﬁnishes early ] = 0.84

orchestra answered 1 year ago

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