a poll classified 74 respondents by gender and political par

a poll classified 74 respondents by gender and political party, as shown. Find the probabilities. Write answers as simplified fractions or as a decimal numbers rounded to four decimal places. Males 13 democrats, 14 republicans, 7 independent for a total of 34. Females 7 democrats, 23 republicans, 10 independents total of 40. Totals counts = 20 democrats, 37 republicans, 17 independents. Males = 34, females = 40 grand total = 74. A) What is the probability that a randomly selected individual is both republican and female? B) What is the probability that a randomly selected person is a Democrat, given that the person is Female? C) What is the probability that a randomly selected person is a male or a democrat. D) If two people are selected at random, find the probability that both are republicans. E) If two people are selected at random, find the probabiity that at least one is a democrat. F) Are being a female and being a democrat independent events? G) Are being a female and being a democrat mutually exclusive events?

Solution

probability that a randomly selected individual is both republican and female = 23/74 = 0.3108

probability that a randomly selected person is a Democrat, given that the person is Female = 7/40 = 0.175

probability that a randomly selected person is a male or a democrat = (34 + 20 - 13)/74 = 41/74 = 0.5541

If two people are selected at random, the probability that both are republicans = (37/74)*(36/73) = 0.2466

If two people are selected at random, the probabiity that at least one is a democrat = 1 - (54/74)*(53/73) = 0.4702

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Are being a female and being a democrat independent events?

P(female) = 40/74

P(democrat) = 20/74

P(female and demcrat) = 7/74 = 0.0946

P(female) * P(democrat) = 0.1461

As, They are not same, being a female and being a democrat are not independent events

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Are being a female and being a democrat mutually exclusive events?

P(Feamle and democrat) = 7/74

As, P(Feamle and democrat) is not equal to 0, being a female and being a democrat are not mutually exclusive events