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.

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

(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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