18. Suppose that we asked 2000 people what their blood type is (O, A, Bor AB) an
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Question
18. Suppose that we asked 2000 people what their blood type is (O, A, Bor AB) and what ethnic group they belong to (G1, G2 or G3). The results of the survey are listed in the table below Blood Type Total Ethnic Group G1 G2 G3 Total 0 300 125 165 590 150 650 145 945 175 15 175 365 AB 25 10 65 100 2000 650 800 550 The numbers in the center of the table are the number of the people in the survey that answered the ethnic question as stated in that row and answered the blood type question as stated in that column. For example, 300 people have type O blood and are in group G1. The totals are in the outer row and column of the table. The total of 2000 can be determined by summing up the row totals or summing up the column totals. Determine the following probabilities. a) P(G1) b) P(A) c) P(AUB) j) P(Bc)Explanation / Answer
a)
P(G1) = Total number of people in ethnic group G1 / Total number of people surveyed
Total number of people in ethnic group G1 = 650
Total number of people surveyed = 2000
P(G1) = 650/2000 = 0.325
b)
P(A) = Total number of people having blood group A / Total number of people surveyed
Total number of people having blood group A = 945
Total number of people surveyed = 2000
P(A) = 945/2000 = 0.4725
c)
P(A U B) = Number of people having blood group 'A' or 'B' / Total number of people surveyed
Since A and B are mutually exclusive events P(AB) = 0
P(AUB) = (945+365) / 2000 = 0.655
d)
P(AG2) = Number of people having blood group 'A' and belonging to ethnic group 'G2' / Total number of people surveyed
P(AG2) = 650/2000 = 0.325
e)
P(G1 U G2) = Number of people belonging to ethnic group 'G1' or 'G2' / Total number of people surveyed
P(G1 U G2) = (650 + 800) / 2000 = 0.725
f)
P(G3 O) = Number of people having blood group 'O' and belonging to ethnic group 'G3' / Total number of people surveyed
P(G3 O) = 165 / 2000 = 0.0825
g)
P(G3 U O) = P(G3) + P(O) - P(G3O)
P(G3) = 550/2000 = 0.275
P(O) = 590/2000 = 0.295
P(G3 O) = 165 / 2000 = 0.0825
P(G3 U O) = 0.275 + 0.295 - 0.0825 = 0.4875
h)
P(G1 U G2 U AB) = P[(G1 U G2) U AB] = P(G1 U G2) + P(AB) - P[(G1 U G2) AB]
P(G1 U G2) = (650+800) / 2000 = 0.725
P(AB) = 100/2000 = 0.05
P[(G1 U G2) AB] = 0.0175 (see part (i))
P(G1 U G2 U AB) = 0.725 + 0.05 - 0.0175 = 0.7575
i)
P[(G1 U G2) AB] = Number of people belonging to either 'G1' or 'G2' and having blood group AB / Total number of people surveyed
Number of people belonging to either 'G1' or 'G2' and having blood group AB = 25+10 = 35
P[(G1 U G2) AB] = 35/2000 = 0.0175
j)
P(Bc) = 1 - P(B)
P(B) = 365/2000 = 0.1825
P(Bc) = 0.8175
k)
P[(A U G3)c] = 1 - P(A U G3)
P(A U G3) = P(A) + P(G3) - P(AG3)
P(A) = 945/2000 = 0.4725
P(G3) = 550/2000 = 0.275
P(AG3) = 145/2000 = 0.0725
P(A U G3) = 0.4725 + 0.275 - 0.0725 = 0.675
P[(A U G3)c] = 1 - 0.675 = 0.325
l)
P[(G2 AB)c] = 1 - P(G2 AB)
P(G2 AB) = 10/2000 = 0.005
P[(G2 AB)c] = 1 - 0.005 = 0.995
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