Chapter 13.2, Problem 4CYU

(a)

To determine

To find:the probability that the arrangement shown at the right is produced.

Answer to Problem 4CYU

The probability is $\frac{1}{120}$ .

Explanation of Solution

Given:

In chemistry lab need to test six samples.

Concept used:

The number of ways of arranging $n$ item in order $=n!$ .

The probability formula is used to compute the probability of an event to occur as:

  $P\left(A\right)=\frac{\text{Number of favorable outcome}}{\text{Total Number of favorable outcomes}}$ .

Calculation:

The number of distinguishable permutations of  $n$ objects arranged in a circle with no fixed reference point is:

  $\frac{n!}{n}=\left(n-1\right)!$ .

the  $6$  samples can be arranged in $5!=120$ different ways. Therefore, the probability is

  $\frac{1}{120}$ .

Hence, the probability is $\frac{1}{120}$ .

(b)

To determine

To find: the probability that the test tube $2$ will be in the top middle position.

Answer to Problem 4CYU

The probability is $\frac{1}{6}$ .

Explanation of Solution

Given:

In chemistry lab need to test six samples.

Concept used:

The number of ways of arranging $n$ item in order $=n!$ .

The probability formula is used to compute the probability of an event to occur as:

  $P\left(A\right)=\frac{\text{Number of favorable outcome}}{\text{Total Number of favorable outcomes}}$ .

Calculation:

Since, the samples are arranged with a fixed reference point. This is a linear permutation. So, there are $6!$ or $720$ ways in which the samples can be arranged. The number of favourable outcomes is the number of permutations of the other $5$ samples given that the test tube $2$ will be in the top middle position. $5!$ or $120$ . Therefore, the probability is $\frac{120}{720}=\frac{1}{6}$ .

Hence, the probability is $\frac{1}{6}$ .