Probability with random?

In a survey of 1000 eligible voters selected at random, it was found that 62% had a college degree.





Additionally, the survey results showed tha of the eligible voters who had a college degree, 85% voted in the most recent senatorial election.





Furthermore, only 42% of the eligible voters without a college degree voted in the same election.





Based on this information and assuming it is representative of the general voting population, find the probability that a


randomly selected eligible voter





a. voted in the last senatorial election.





b. has a college degree and did not vote in the last senatorial election.





c. does not have a college degree and did not vote in the last senatorial election.





How can I solve this? :-|


Thanks.

Probability with random?
a. Separate the eligible voters into those who have a college degree and those who do not by the probability of each. Since 62% of them have a degree, 38% do not.





Degree: 62/100


No Degree: 38/100





Now, calculate the probabilities of a person having a degree and voting in the election, and not having a degree and voting in the election separately.





Degree: (62/100) * (85/100) = 5270/10000


No Degree: (38/100) * (42/100) = 1596/10000





Sum these two values, and you have the probability of a person being in either group and voting in the election.





Voting: (5270/10000) + (1596/10000) = 6866/10000 = 3433/5000. As a percentage, 68.66%.








b. We already know that 62% of eligible voters have a college degree. Now we need to calculate the probability of having a college degree and not voting in the election. This is done in a similar manner as step two of part a.





Since 85% of people with a college degree voted in the last election, it is obvious that 15% of those people did not.





Degree and No Vote: (62/100) * (15/100) = 930/10000 = 93/1000. As a percentage, 9.3%.








c. This is done just like part b, but using the group of voters without degrees. Since 42% of them voted, 58% did not.





No Degree, No Vote: (38/100) * (58/100) = 2204/10000 = 551/2500. As a percentage, 22.04%.
Reply:a. 68.66%


b. 9.3%


c. 22.04%





calculated


a.


(62% x 85%) = 52.7% (percentage that have degree and voted)


38% x 42% = 15.96% (percentage that have no college degree and voted)





52.7%+15.96% = 68.66%





b.


62% (have college degree) x 15% ( percentage not voting) = 9.3%





c.


38% ( no college degree) x 58% ( percent not voting) = 22.04%
Reply:Let


P(c) = probability of having college degree


P(v) = probability of voting in last election





Given





P(c) = 0.62


P(v|c) = 0.85


P(v|~c) = 0.42





a) Calculate P(v).





P(~c) = 1 - P(c) = 1 - 0.62 = 0.38





P(v) = P(v|c)P(c) + P(v|~c)P(~c) = 0.85*0.62 + 0.42*0.38


P(v) = 0.6866


________________





b) Calculate P(c∩~v).





P(~v|c) = 1 - P(v|c) = 1 - 0.85 = 0.15





P(c∩~v) = P(~v|c)P(c) = 0.15*0.62 = 0.093


_______________





c) Calculate P(~c∩~v).





P(~v|~c) = 1 - P(v|~c) = 1 - 0.42 = 0.58





P(~c∩~v) = P(~v|~c)P(~c) = 0.58*0.38 = 0.2204
Reply:take event C as voter had a college degree,


and event E as he/she participate in recent election.


^ : intersection


U : union


| : conditional


' : compliment / negation





P(C) = 0.62


P(C | E) = 0.85


P(E^C') = 0.42 ---%26gt; P(E ^ C) = 1 - 0.42 = 0.58





a. Probability that a randomly voter vote,


P(E) = ...?


*


Bayes theorem:


P(C | E) = P( E ^ C) / P(E)


P(E) = P(E ^ C) / P(C | E) = 0.58 / 0.85


= 0.682352941


Expected Value of E :


Mean(E) = 1000 * 0.682 = 682


.: 682 eligible voters voted in last election





b. has college degree and did note vote:


P(C^E') = ...?


**


Assume events C and E are uncorrelated and so independent


P(C^E) = P(C) * P(E) = 0.62 * (1-0.682352941)


= 0.196941177


Expected value of C^E:


Mean (C^E') = 0.197 * 1000


= 197


.: 197 college graduates did not vote





c. doesn't have college degree nor vote:


P(C'^E') = ...?





P(C'^E') = P(C') * P(E') = (1-0.62) * (1-0.682352941)


= 0.120705882


Expected vale\ue of C'^E':


Mean (C'^E') = 0.121 * 1000 = 121


.: 121 alligibles who did not vot don't have any college degree





You can learn more from





Probability %26amp; Statistics for Engineers %26amp; Scientists written by Ronald E. Walpole.


Probability and Statistics written by Athanasios Papoulis

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