An academic department has 11 faculty members 6 of whom are...

An academic department has 11 faculty members, 6 of whom are tenured. Five faculty members are to be randomly selected to serve on a committee. Let X be a random variable for the number of tenured members on the committee. (a) Specify the distribution for X, and give the parameters of the distribution. (b) What is the probability that the tenured members on the committee will constitute a majority? (c) Compute the mean and variance for the number of tenured members on the committee.

Transcript

00:01 This question is saying that we have 11 here faculty members and we have the six of these members here they are tenured so basically this is what we are interested in okay so in this question basically are only interested in the if you want to select for example five people here from this group of 11 we want to know how many of them are tenor here.

00:38 So because we are counting this, and we know the specific number of people that are being selected, and we know how many we have from the entire population of faculty members to select from, this means the x here, the distribution is binomial.

00:56 And the parameters here is the number of people being selected, five, and the probability of getting a specific, like in this case getting one that is like 10 out of the 11 that we have.

01:09 So for example, we have six out of 11.

01:14 They have this specific characteristic that we are interested in.

01:19 So this is what we call the probability for the binomial distribution.

01:23 This is the other parameter.

01:25 Now in item b, we should use this to find what is the probability that the majority of the members that they select will be tenured.

01:34 So basically, this means that if you're selecting five, we are going to have the majority if you select at least three.

01:42 So basically, we want to compute this, x being greater or equal than three.

01:48 But when we say this, we know that at maximum we are selecting five, so we can list the options that we have.

01:55 So we can have x equals to three, x equals to four, x equals to five.

02:01 And for each one of this, we can use the binomial distribution to compute this.

02:07 So here to compute for x equals to 3, we are going to use the formula given by the binomal distribution.

02:12 Which first we put the combination of n and the number that we have here, 3...

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