Chapter 12, Problem AP4.44CPT

(a)

To determine

To find out what is the probability that on a randomly selected morning the coffee will be strong.

Answer to Problem AP4.44CPT

The probability that on a randomly selected morning the coffee will be strong is $0.16$ .

Explanation of Solution

It is given that each morning, coffee is brewed in the school workroom by one of the three faculty members depending on who arrives at first for work. Also,

  $\begin{array}{l}P(\text{Mr}\text{. Worcester)}=0.10\\ P(\text{Dr}\text{. Currier)}=0.50\\ P(\text{Strong coffee | Dr}\text{. Currier)}=0.1\\ P(\text{Strong coffee | Mr}\text{. Worcester})=0.3\\ P(\text{Strong coffee | Mr}\text{. Legacy)}=0.2\end{array}$

Now, we will find the probability for Mr. Legacy and that will be the remaining days other than the two. Thus, we have,

  $\begin{array}{l}P(\text{Mr}\text{. Legacy})=1-0.1-0.5\\ =0.4\\ =40\%\end{array}$

Now, using the multiplication rule, we have,

  $\begin{array}{l}P(\text{Strong coffee and Dr}\text{. Currier)}=P(\text{Dr}\text{. Currier)}\times P(\text{Strong coffee | Dr}\text{. Currier)}\\ =0.50\times 0.1\\ =0.05\\ P(\text{Strong coffee and Mr}\text{. Worcester)}=P(\text{Mr}\text{. Worcester)}\times P(\text{Strong coffee | Mr}\text{. Worcester})\\ =0.1\times 0.3\\ =0.03\\ P(\text{Strong coffee and Mr}\text{. Legacy)}=P(\text{Mr}\text{. Legacy})\times P(\text{Strong coffee | Mr}\text{. Legacy)}\\ =0.4\times 0.2\\ =0.08\end{array}$

Now, we will find the probability that on a randomly selected morning the coffee will be strong by using the addition rule,

  $\begin{array}{l}P(\text{Strong coffee})\\ =P(\text{Strong coffee and Dr}\text{. Currier)}+P(\text{Strong coffee and Mr}\text{. Worcester)}+P(\text{Strong coffee and Mr}\text{. Legacy)}\\ =0.05+0.03+0.08\\ =0.16\\ =16\%\end{array}$

Thus, the probability that on a randomly selected morning the coffee will be strongis $0.16$ .

(b)

To determine

To explain what is the probability that it was brewed by Mr. Currier if the coffee is strong ona randomly selected morning.

Answer to Problem AP4.44CPT

The probability that it was brewed by Mr. Currier if the coffee is strong on a randomly selected morning is $31.25\%$ .

Explanation of Solution

It is given that each morning, coffee is brewed in the school workroom by one of the three faculty members depending on who arrives at first for work. Also,

  $\begin{array}{l}P(\text{Mr}\text{. Worcester)}=0.10\\ P(\text{Dr}\text{. Currier)}=0.50\\ P(\text{Strong coffee | Dr}\text{. Currier)}=0.1\\ P(\text{Strong coffee | Mr}\text{. Worcester})=0.3\\ P(\text{Strong coffee | Mr}\text{. Legacy)}=0.2\end{array}$

And from the above part (a),

  $\begin{array}{l}P(\text{Mr}\text{. Legacy})=1-0.1-0.5\\ =0.4\\ =40\%\end{array}$

  $\begin{array}{l}P(\text{Strong coffee and Dr}\text{. Currier)}=P(\text{Dr}\text{. Currier)}\times P(\text{Strong coffee | Dr}\text{. Currier)}\\ =0.50\times 0.1\\ =0.05\\ P(\text{Strong coffee and Mr}\text{. Worcester)}=P(\text{Mr}\text{. Worcester)}\times P(\text{Strong coffee | Mr}\text{. Worcester})\\ =0.1\times 0.3\\ =0.03\\ P(\text{Strong coffee and Mr}\text{. Legacy)}=P(\text{Mr}\text{. Legacy})\times P(\text{Strong coffee | Mr}\text{. Legacy)}\\ =0.4\times 0.2\\ =0.08\end{array}$

And the probability that on a randomly selected morning the coffee will be strong is $0.16$ .

Thus,

  $\begin{array}{l}P(\text{Dr}\text{. Currier | Strong coffee)}=\frac{P(\text{Strong coffee and Dr}\text{. Currier)}}{P(\text{Strong coffee})}\\ =\frac{0.05}{0.16}\\ =0.3125\\ =31.25\%\end{array}$

Thus, the probability that it was brewed by Mr. Currier if the coffee is strong on a randomly selected morningis $31.25\%$ .