Question

The only contents of a container are 10 disks that are each numbered with a different positive integer from 1 through 10, inclusive. if 4 disks are to be selected one after the other, with each disk selected at random and without replacement, what is the probability that the range of the numbers on the disks selected is 7

Answer 1

First we calculate for the total number of possibilities (permutation) to select 4 disks from the container:

Total number of possibilities = 10 * 9 * 8 * 7

Total number of possibilities = 5040

Now let us find the 4 disks that will result in a range of 7.

Range = highest number – lowest number

The pair of highest and lowest number that will result in range of 7 is: (1 & 8), (2 & 9), (3 & 10)

 

As a basis of calculation, let us use the pair 1 & 8.

There are four possible ways to select 1 and three for 8.

Arrangements of maximum and minimum pair = 4 * 3

Arrangements of maximum and minimum pair=12

Now we need to calculate for the remaining 2 disk. There are 6 numbers between 1 & 8. The total possibilities for selecting 2 disk from the remaining 6 is:

Possibilities of selecting 2 disk from remaining 6 = 6 * 5

Possibilities of selecting 2 disk from remaining 6 = 30

Therefore, the total possibility to get a range of 7 from a pair of 1 & 8 is:

Total possibility for a pair = 12 * 30

Total possibility for a pair = 360

 

Since there are a total of three pairs (1 & 8), (2 & 9), (3 & 10):

Total possibilities of the 3 pairs = 360 * 3

Total possibilities of the 3 pairs = 1080

 

Therefore:

Probability = Total possibilities of the 3 pairs / Total number of possibilities

Probability = 1080 / 5040 = 3 / 14                      (FINAL ANSWER)