This document summarizes data on US Senators from the 112th Congress whose terms expire in 2013, 2015, or 2017. It provides the numbers of Democratic and Republican Senators in each expiration year, and calculates the probabilities of various events based on this data. Specifically, it calculates: (a) the probability a randomly selected Senator is Democratic and their term expires in 2015, (b) the probability a randomly selected Senator is Republican or their term expires in 2013, and (c) the probability a Senator is Republican given their term expires in 2017. It also determines whether being Republican and having one's term expire in 2015 are independent events.

1. Below is given the summary from the 112th Congress of Senators whose terms end
in 2013, 2015, or 2017.
2013 2015 2017 TOTAL
Democrat 21 20 1 42
Republican 8 15 13 36
Total 29 35 14 78
Chooseone of these Senators at random and find
(a) P(Democrat and term expires in 2015)
(b) P(Republican or term expires in 2013)
(c) P(Republican given term expires in 2017)
(d) Are the events “Republican” and “term expires in 2015” independent?
Explain.
Solution
Let
A = Term expires in 2013
B = Term expires in 2015
C = Term expires in 2017
D = Democrat
R = Republican
𝑃(𝐴) =
29
78
𝑃(𝐵) =
35
78
𝑃(𝐶) =
14
78

2. 𝑃(𝐷) =
42
78
𝑃(𝑅) =
36
78
(a) 𝑃(𝐷 𝑎𝑛𝑑 Term expires in 2015) = 𝑃(𝐷 ∩ 𝐵) =
𝟐𝟎
78
(b)𝑃(𝑅 𝑜𝑟 Term expires in 2013) = 𝑃(𝑅 ∪ 𝐴) = 𝑃(𝑅) + 𝑃(𝐴) − 𝑃(𝑅 ∩ 𝐴)
𝑃(𝑅 ∪ 𝐴) =
𝟑𝟔
78
+
𝟐𝟗
78
−
𝟖
78
= 0.7308
(c)𝑃(𝑅
𝐶
⁄ ) =
𝑃(𝑅∩𝐶)
𝑃(𝐶)
=
𝟏𝟑
78
⁄
𝟏𝟒
78
⁄
=
13
14
= 0.9286
(d)If events are Independent then
𝑃(𝑅 ∩ 𝐵) = 𝑃(𝑅) × 𝑃(𝐵)
𝐵𝑢𝑡
15
78
≠
36
78
×
35
78
Therefore events are not independent.