Question

A 15-member Board of Directors is organizing for the upcoming year.   

2.         They must elect a president, a vice-president, and a treasurer.  How many different slates of candidates are possible?  

3.         They must choose four members to serve on the finance committee. How many different committees could be formed?

The age and gender breakdown of the Board is

	Female	Male
25 – 35 years old	3	2
36 – 45 years old	1	4
Over 46 years old	2	3

If one member is randomly chosen to organize the annual meeting, what is the probability that the person will be

4.         Female

            

5.         A female or a person 25-35 years old

6.         A male between 36 and 45 years old.

The Board has narrowed the options for the annual meeting to the following:

City:  Annapolis or Baltimore

Month:  March or June or September  

Venue: Hotel or College

7.         Draw a tree diagram to show the sample space for all possible choices.  

8.         If one package is chosen at random what is the probability that it is in Annapolis at a hotel?

Answer 1

Please note ⁿC_x = n! / [(n-x)!*x!]

Some important probabilities for faster calculations:

(a) ⁿC₀ = 1, eg⁵C₀ = 1, ⁹⁹C₀ = 1

(b) ⁿC_n = 1, eg¹⁰C₁₀ = 1, ⁷⁵C₇₅ = 1

(c) ⁿC₁ = n, ,eg⁵C₁ = 5, ⁹⁹C₁ = 99

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2) Since the president, VP and treasurer cannot hold 2 posts at the same time, the total possible ways

= ¹⁵C₁ * ¹⁴C₁ * ¹³C₁

= 15 * 14 * 13 = 2730

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3) 4 Members for the finance committee. Choose 4 out of 15 = ¹⁵C₄= 1365

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For Questions 4, 5 and 6

Probability = Favorable Outcomes / Total Outcomes

	Female	Male	Total
25 – 35 years old	3	2	5
36 – 45 years old	1	4	5
Over 46 years old	2	3	5
Total	6	9	15

4) P(That a Randomly chosen person is a Female) = 6/15 = 2/5 = 0.4

5) P(A Female or a Person 25 - 35 Years Old) = P(Female) + P(A person 25 - 35 years old) - P(A Female and a Person 25 - 35 Years Old)

= 6/15 + 5/15 - 3/15 = 8/15 = 0.533

6) (A Male between 36 - 45 years old) = 4/15 = 0.267

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8) P(Annapolis at a Hotel): By referring to the diagram, the required probability = 1/4

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7) The Tree Diagram is as below. The Individual Probabilities are

P(Annapolis) = P(Baltimore) = 1/2

P(March) = P(June) = P(September) = 1/3

P(Hotel) = P(College) = 1/2