Question 1160215: Urgent Help Needed
Below is given the summary from the 112th Congress of Senators whose terms end in 2013, 2015, or 2017.
2013 2015 2017
Democrat 21 20 1
Republican 8 15 13
Choose one 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.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the following concepts apply.
p(a or b) = p(a) + p(b) - p(a and b).
p(a and b) = p(a) * p(b) if the events are independent.
p(a given b) = p(a and b) / p(b).
p(a given b) = p(a0 if the events are independent.
if you draw a table of what you are given, you would find the table looks like this:
politics 2013 2015 2017 total
democrat 21 21 1 42
republican 8 15 13 36
total 29 36 14 78
let d be the event that the person is democrat.
let r be the event that the person is republican.
let 2013 be the event that the person retires in 2013.
let 2015 be the event that the person retires in 2015.
let 2017 be the event that the person retires in 2017.
from the table:
p(d) = 42 / 78 because there are 42 democrats out of a total of 78 senators who retired in either 2013 or 2015 or 2017.
p(r) = 36 / 78 because there are 36 republicans out of a total of 78 senators who retired in either 2013 or 2015 or 2017.
p(2013) = 29 because there are 29 senators who retired in 2013 out of a total of 78 senators who retired in all 3 years
p(2015) = 35 because there are 35 senators who retired in 2015 out of a total of 78 senators who retired in all 3 years.
p(2017) = 14 because there are 14 senators who retired in 2017 out of a total of 78 senators who retired in all 3 years.
your universe is 78 senators who retired in all 3 years of (2013, 2015, 2017).
42 are democrats and 36 are republicans.
29 retired in 2013 and 35 retired in 2015 and 14 retired in 2017.
your questions are:
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.
answer to a):
from the table, you can see that there are 20 democrats who retired in 2015.
therefore p(d and 2015) = 20/78.
answer to b):
this one is asking for p(r or 2013)
the formula to use here is p(r or 2013) = p(r) + p(2013) - p(r and 2013).
from the table, you can see that:
p(r) = 36/78
p(2013) = 29/78
p(r and 2013) = 8
therefore:
p(r or 2013) is equal to p(r) + p(2013) - p(r and 2013) which is equal to 36/78 + 29/78 - 8/78 = 57/78.
answer to c):
this is one is asking for p(r given 2017)
the formula to use here is p(r given 2017) = p(r and 2017) / p(2017)
from the table, you can see that:
p(2017) = 14/78
p(r and 2017) = 13/78
therefore:
p(r given 2017) = p(r and 2017) / p(2017) = (13/78) / (14/78) = 13/14.
answer to d):
it they are independent, then p(r given 2015) would be equal to p(r).
why is this so?
if they were independent, then p(r and 2017) would be equal to p(r) * p(2017).
therefore p(r given 2017) would be equal to p(r and 2017) / p(2017) which would be equal to p(r) * p(2017) / p(2017) which would be equal to p(r).
from the table, you can see that p(r) = 36/78.
from the table, you can see that p(r and 2017) = 13/78) and p(2017) = 14/78.
therefore p(r given 2017) = p(r and 2017) / p(2017) = (13/78) / (14/78) which is equal to 13/14 which is not equal to p(r) = 36/78
therefore the two events are not independent.
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