Chapter 12, Problem 12.50E

a.

To determine

To find: All possible arrangements for selecting the three cups from six cups.

Answer to Problem 12.50E

The all possible arrangements for selecting three cups from six cups

$S=\left\{\begin{array}{l}\left(M_1{M}_2{T}_1\right),\left(M_1{M}_2{T}_2\right),\left(M_1{M}_2{T}_3\right),\left(M_1{M}_3{T}_1\right),\left(M_1{M}_3{T}_2\right),\left(M_1{M}_3{T}_3\right)\\ \left(M_2{M}_3{T}_1\right),\left(M_2{M}_3{T}_2\right),\left(M_2{M}_3{T}_3\right),\left(T_1{T}_2{M}_1\right),\left(T_1{T}_2{M}_2\right),\left(T_1{T}_2{M}_3\right)\\ \left(T_1{T}_3{M}_1\right),\left(T_1{T}_3{M}_2\right),\left(T_1{T}_3{M}_3\right),\left(T_2{T}_3{M}_1\right),\left(T_2{T}_3{M}_2\right),\left(T_2{T}_3{M}_3\right)\\ \left(M_1{M}_2{M}_3\right),\left(T_1{T}_2{T}_3\right)\end{array}\right\}$

Explanation of Solution

Given info:

A Canadian friend, who is fond of tea, claims that it is possible to differentiate the taste of tea. While preparing tea, milk is added first and then hot tea is poured. The other way is hot tea is added first and then milk is poured. To prove the claim of Canadian friend, six cups of tea were prepared. Three of the cups were prepared using the first method and other three cups were prepared by using the second method. In a test, the friend is asked to find the three cups which had milk first and then hot tea poured into it.

Justification:

Sample space:

The sample space is defined as the set of all possible outcomes from an experiment.

There are six cups of tea, first three cups have milk added first and then hot tea poured into it. The remaining three cups have hot tea added first and then milk is poured into it.

Define $M_1$ as the milk added first in the first cup.

Define $M_2$ as the milk added first in the second cup.

Define $M_3$ as the milk added first in the third cup.

Define $T_1$ as the hot tea added first in the first cup.

Define $T_2$ as the hot tea added first in the second cup.

Define $T_3$ as the hot tea added first in the third cup.

It is mentioned that the three cups were first added with milk and then hot tea. Another three cups were first added with hot tea and then milk.

One way to select the three cups from the given six cups can be of milk first added in the first cup, milk first added in the second cup and hot tea first added in the first cup. That is, $M_1{M}_2{T}_1$. Similarly, other outcomes can be obtained.

Thus, the possible arrangements for selecting three cups from six cups are,

$S=\left\{\begin{array}{l}\left(M_1{M}_2{T}_1\right),\left(M_1{M}_2{T}_2\right),\left(M_1{M}_2{T}_3\right),\left(M_1{M}_3{T}_1\right),\left(M_1{M}_3{T}_2\right),\left(M_1{M}_3{T}_3\right)\\ \left(M_2{M}_3{T}_1\right),\left(M_2{M}_3{T}_2\right),\left(M_2{M}_3{T}_3\right),\left(T_1{T}_2{M}_1\right),\left(T_1{T}_2{M}_2\right),\left(T_1{T}_2{M}_3\right)\\ \left(T_1{T}_3{M}_1\right),\left(T_1{T}_3{M}_2\right),\left(T_1{T}_3{M}_3\right),\left(T_2{T}_3{M}_1\right),\left(T_2{T}_3{M}_2\right),\left(T_2{T}_3{M}_3\right)\\ \left(M_1{M}_2{M}_3\right),\left(T_1{T}_2{T}_3\right)\end{array}\right\}$

Thus, there are 20 possible selections to select three from six cups.

b.

To determine

To find: The probability that the Canadian friend identifies the three cups in which milk is added first.

Answer to Problem 12.50E

The probability that the Canadian friend is identifying the three cups in which milk is added first is $\frac{1}{20}$.

Explanation of Solution

Given info:

Assume that the Canadian friend is just making a random guess.

Calculation:

The probability that the Canadian friend identifies the three cups in which milk is added first is calculated as follows:

The number of ways in which milk is added first in three cups are $\left(M_1{M}_2{M}_3\right)$.

$\begin{array}{c}\left(\begin{array}{l}\text{Probability of selecting}\\ \text{three cups in which }\\ \text{milk is added first }\end{array}\right)=\frac{\text{Number of ways in which milk is added first in three cups}}{\text{Total number of ways to select three cups}}\\ =\frac{1}{20}\end{array}$

Thus, the probability that the Canadian friend is identifying the three cups in which milk is added first is $\frac{1}{20}$.