b If n objects are arranged in a circle relative to a fixed point of reference then the arrangements are treated as linear, making the number of permutations n!.
A
a 1/120
B
b 1/6
Practice makes perfect
a Six samples are randomly arranged on a circular tray. We are asked to find the probability that the following arrangement is produced.
Notice that there is no fixed reference point, so we can rotate the tray however we want and the arrangement is still considered the same. Therefore, the number of distinguishable arrangements of the probes can be calculated using the Circular Permutations Formula.
Circular Permutations
The number of distinguishable permutations of n objects arranged in a circle with no fixed point of reference can be calculated using one of the following formulas.
n!/n or( n-1)!
There are 6 probes on the tray, so let's substitute n= 6 into one of the formulas!
The number of distinguishable permutations of the probes is 120. Therefore, the probability that the given arrangement is produced is 1 120.
b We are asked to find the probability that test tube 2 will be in the top middle position.
Notice that the top middle tube spot is now a fixed reference point. Rotating the trace changes which tube is in the top middle tube spot. Therefore, we can treat this as a linear permutation. There are 6 tube spots, thus the total number of distinguishable linear permutations of the tubes is 6!.
Possible outcomes= 6!
Now, let's calculate the number of favorable outcomes, which is the number of the permutations of the tubes in which the top middle tube spot is taken by tube number 2.
This number is equal to the number of linear permutations of the other 5 tubes. Therefore, the number of favorable outcomes is equal to 5!. Finally, we can calculate the probability that tube number 2 is in the top middle spot by dividing the number of favorable outcomes by the number of possible outcomes.
P=Favorable outcomes/Possible outcomes=5!/6!
Let's calculate the quotient!
