Chapter 12: Q. 44 (page 807)

Each morning, coffee is brewed in the school workroom by one of three faculty members, depending on who arrives first at work. Mr. Worcester arrives first 10% of the time, Dr. Currier arrives first $50 %$ of the time, and Mr. Legacy arrives first on the remaining mornings. The probability that the coffee is strong when brewed by Dr. Currier is 0.1, while the corresponding probabilities when it is brewed by Mr. Legacy and Mr. Worcester are $0.2$ and $0.3$ , respectively. Mr. Worcester likes strong coffee!
(a) What is the probability that on a randomly selected morning the coffee will be strong? Show your work.
(b) If the coffee is strong on a randomly selected morning, what is the probability that it was brewed by Dr. Currier? Show your work.

Short Answer

Expert verified

(a) The probability that on a randomly selected morning the coffee will be strong is $16 %$ .

(b) The probability that it was brewed by Dr. Currier is $31.25 %$ .

Step by step solution

Part (a) step 1: Given Information

We need to find the probability that on a randomly selected morning the coffee will be strong

Part (a) step 2: Simplify

Multiplication rule:

$P (A a n d B) = P (A \cap B) = P (A) \times P (B ∣ A) = P (B) \times P (A ∣ B)$

Addition rule for disjoint or mutually exclusive events:

$P (A \cup B) = P (A o r B) = P (A) + P (B)$

$P (M r . W o r c e s t e r) = 10 % = 0.10$

localid="1650619178904" $P (D r . C u r r i e r) = 50 % = 0.50$

$P (S t r o n g c o f f e e D r . C u r r i e r) = 0.1 P (S t r o n g c o f f e e ∣ M r . W o r c e s t e r) = 0.3 P (S t r o n g c o f f e e ∣ M r . L e g a c y) = 0.2$

Mr. Worcester arrives first $10 %$ of the time and Dr. Currier arrives first $50 %$ ,so the time, while Mr. Legacy arrives first on the remaining days and this is then $100 % - 50 % - 10 % = 40 %$ of the time:

localid="1650619365441" $P (Mr . L e g a c y) = 40 % = 0.40$

Use the general multiplication rule is :

localid="1650619862930" $P (S t r o n g c o f f e e a n d D r . C u r r i e r) = P (D r . C u r r i e r) \times P (S t r o n g c o f f e e ∣ D r . C u r r i e r) = 0.50 \times 0.1 = 0.05 P (S t r o n g c o f f e e a n d M r . W o r c e s t e r) = P (M r . W o r c e s t e r) \times P (S t r o n g c o f f e e ∣ M r . W o r c e) = 0.10 \times 0.3 = 0.03 P (s t r o n g c o f f e e a n d M r . L e g a c y) = P (M r . L e g a c y) \times P (s t r o n g c o f f e e ∣ M r . L e g a c y) = 0.40 \times 0.2 = 0.08$

Now, the addition rule of disjoint events:

$P (S t r o n g c o f f e e) = P (S t r o n g c o f f e e a n d D r . C u r r i e r) \times P (S t r o n g c o f f e e a n d M r . W o r c e s t e r) + P (S t r o n g c o f f e e a n d M r . Legacy) = 0.5 + 0.3 + 0.8 = 0.16 = 16 %$

Part (b) step 1: Given Information

We need to find the probability that it was brewed by Dr. Currier.

Part (b) step 2: Simplify

Multiplication rule:

$P (A a n d B) = P (A \cap B) = P (A) \times P (B ∣ A) = P (B) \times P (A ∣ B)$

Addition rule for disjoint or mutually exclusive events:

$P (A \cup B) = P (A o r B) = P (A) + P (B)$

$P (M r . W o r c e s t e r) = 10 % = 0.10$

$P (D r . C u r r i e r) = 50 % = 0.50$

$P (S t r o n g c o f f e e D r . C u r r i e r) = 0.1 P (S t r o n g c o f f e e ∣ M r . W o r c e s t e r) = 0.3 P (S t r o n g c o f f e e ∣ M r . L e g a c y) = 0.2$

$P (Mr . L e g a c y) = 40 % = 0.40$

Use the general multiplication rule is :

$P (S t r o n g c o f f e e a n d D r . C u r r i e r) = P (D r . C u r r i e r) \times P (S t r o n g c o f f e e ∣ D r . C u r r i e r) = 0.50 \times 0.1 = 0.05 P (S t r o n g c o f f e e a n d M r . W o r c e s t e r) = P (M r . W o r c e s t e r) \times P (S t r o n g c o f f e e ∣ M r . W o r c e) = 0.10 \times 0.3 = 0.03 P (s t r o n g c o f f e e a n d M r . L e g a c y) = P (M r . L e g a c y) \times P (s t r o n g c o f f e e ∣ M r . L e g a c y) = 0.40 \times 0.2 = 0.08$

Now, the addition rule of disjoint events:

Use the definition of conditional probability:

$P (D r . C u r r i e r | S t r o n g c o f f e e) = \frac{P (S t r o n g c o f f e e a n d D r . C u r r i e r)}{P (S t r o n g c o f f e e)} = \frac{0.05}{0.16} = \frac{5}{16} = 0.3125 = 31.25 %$