The board of directors for a particular company consists of 10 members, 6 of whom are loyal to the current company president and 4 of whom want to fire the president. Suppose the chair of the board (who is a loyal supporter of the current president) suggests to randomly select 4 other board members to serve on a committee to decide the president's fate. There are 4 questions following the table below.

Find the probability for the first 3 questions and explain your answer for the fourth question.

The relevant equation for the Hypergeometric Distribution: P = _kC_x * _(N-k)C_(n-x)

_NC_n

k is the number "loyal" in the population

x is the number "loyal" in the sample

N is the size of the population

n is the number sampled

P is the probability of obtaining exactly x "loyal"

_kC_x is the number of combinations of k things taken x at a time or k!

(k-x)! x!

The table below is constructed using Excel to compute probabilities for obtaining exactly x "loyal" given the conditions above.

k	x	N	n	N-k	n-x	kCx	(N-k)C(n-x)	NCn	P
5	4	9	4	4	0	5	1	126	0.04
5	3	9	4	4	1	10	4	126	0.317
5	2	9	4	4	2	10	6	126	0.476
5	1	9	4	4	3	5	4	126	0.159
5	0	9	4	4	4	1	1	126	0.008

• What is the probability that all 5 committee members will vote to keep the president in place, if no one changes their minds?

• What is the probability that a majority of the committee will vote to keep the president in place, if no one changes their minds?

• What is the probability that the vote is 4 to 1 to replace the president, if no one changes their minds?

• Imagine that you were the president of the company and you hoped to keep your position. Considering the various probabilities, would you consider the chair of the board's suggestion to be in your favor or not? If the choice was yours, would you allow the suggestion to proceed?