Question

In: Statistics and Probability

4. Suppose that 78% of people enjoy drinking coffee and 44% of people enjoy drinking tea....

4. Suppose that 78% of people enjoy drinking coffee and 44% of people enjoy drinking tea. In addition, 39% of people that enjoy drinking coffee also enjoy drinking tea. Let C represent the event that a person enjoys drinking coffee, and T represent the event that a person enjoys drinking tea. In each part of this question, you must first express each probability in terms of the events C and T and justify any computation through the use of a formula.

(a) Express each of the three probabilities listed above in terms of the events C and T.

(b) Are C and T independent events? Explain your answer using only the probabilities from part (a). Do not calculate any other values.

(d) If a person enjoys drinking coffee, what is the probability they do not enjoy drinking tea?

(e) If a person does not enjoy drinking tea, what is the probability they enjoy drinking coffee?

Solutions

Expert Solution

Let C represent the event that a person enjoys drinking coffee, and T represent the event that a person enjoys drinking tea

78% of people enjoy drinking coffee: P(C) = 0.78

44% of people enjoy drinking tea.: P(T) = 0.44

39% of people that enjoy drinking coffee also enjoy drinking tea. This is condictional probability of liking tea given they like coffee: P( T | C) = 0.39

(a) Express each of the three probabilities listed above in terms of the events C and T.

P(C) = 0.78 P(T) = 0.44 P( T | C) = 0.39

(b) Are C and T independent events? Explain your answer using only the probabilities from part (a). Do not calculate any other values.

The events will be independent if

P(C T) =P(T)P(C)

P(C T) / P(C) = P(T)

P( T | C) = P(T) ................P(C T) / P(C) = P(T|C)

P( T | C) = 0.39

P(T) = 0.44

Since P( T | C) P(T)

Events ' C' and 'T' are not independent.

(c) What proportion of the people enjoy drinking both coffee and tea?

.P(C T) / P(C) = P(T|C)

.P(C T) = P(T|C) * P(C)

.P(C T) = 0.3042

(d) If a person enjoys drinking coffee, what is the probability they do not enjoy drinking tea?

This is conditional probability of not drinking tea given enjoys drinking coffee.

P( No T | C) = P(No T C) / P(C)

P(No T C) = P(C) - .P(C T) ..............since with coffee you can either like tea

= 0.4758

P( No T | C) = 0.4758 / 0.78

P( No T | C) = 0.61

(e) If a person does not enjoy drinking tea, what is the probability they enjoy drinking coffee?

This is conditional probability of enjoys drinking coffee given not drinking tea.

P( C | No T) = P(C No T) / P(No T)

P(No T) = 1 - P(T) .............since they are complementary events

= 0.56

P( C | No T) = 0.4758 / 0.56

P( C | No T) = 0.8496

orchestra answered 2 years ago

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