Question

Suppose that for an experimental device setup we have to choose settings for three parameters. There are 4, 3 and 5 settings for parameters 1, 2 and 3 respectively. The order in which the settings will be chosen is not does not matter. How may configurations does the device allow? A random sample of 8 configurations is selected. What is the probability that among the configurations in the sample all three settings for parameter 2 are used.

Answer 1

Using Theorem of counting we know that, since there are 4 settings for parameter 1,3 settings for parameter 2 and 5 settings for parameter 3, thus total configurations that the device allows is (4 x 3 x 5)=60 .

Now for the other part of the question we can consider the settings for parameter 1 and 3 are one entity and that for parameter 2 is another entity.

Therefore, out of 8 samples of configurations ,3 settings that for parameter 2 can be selected in ⁸C₃ ways the probability of the same event happening is (3/12)^3 x (9/12)^5.

3/12 is probability of event of settings for parameter 2 being used and 9/12 is probability of event of settings for parameter 1 and 3 being used(since they were considered as single entity).

Thus, the probability that among the configurations in the sample all three settings for parameter 2 are used=

⁸C₃ x (3/12)^3 x (9/12)^5