Party Affiliation The following data represent political...

Party Affiliation The following data represent political party by age from a random sample of registered Iowa Voters. (a) Are the events "Republican" and "30-44" independent? Justify your answer. (b) Are the events "Democrat" and "65+" independent? Justify your answer. (c) Are the events $^{*} 17-29 "$ and " $45-64 "$ mutually exclusive? Justify your answer. (d) Are the events "Republican" and "45-64" mutually exclusive? Justify your answer.

Party Affiliation The following data represent political party by age from a random sample of registered Iowa Voters.
(a) Are the events "Republican" and "30-44" independent? Justify your answer.
(b) Are the events "Democrat" and "65+" independent? Justify your answer.
(c) Are the events $^{*} 17-29 "$ and " $45-64 "$ mutually exclusive? Justify your answer.
(d) Are the events "Republican" and "45-64" mutually exclusive? Justify your answer.

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Key Concepts

Independence in Probability

Independence in probability refers to a situation where the occurrence of one event does not affect the likelihood of the occurrence of another event. Two events are considered independent if the probability of their intersection equals the product of their individual probabilities, or equivalently, if the conditional probability of one event, given that the other event has occurred, equals its unconditional probability.

Mutually Exclusive Events

Mutually exclusive events are events that cannot occur at the same time; their occurrence is incompatible. This means that if one event occurs, the other event must necessarily not occur, and the probability of their intersection is zero.

Conditional Probability

Conditional probability is the measure of the likelihood of an event occurring given that another event has already occurred. It is essential in determining whether events are independent, as independence can be verified by checking if the conditional probability is equal to the marginal probability.

Contingency Tables

Contingency tables are a type of table in statistics that display the frequency distribution of variables. They are useful for analyzing the relationship between categorical variables and for checking properties like independence or association between events.

Transcript

00:01 So at the top of this problem, i put the rule for independence, which is the probability of event a equals the probability of event a given b.

00:10 If that's the case, then our events are independent.

00:13 So i have the table here.

00:15 The first one's going to ask if the probability of being republican and the probability of being ages 30 to 44 are those two events independent.

00:25 So using the formula from above, if i figure out the probability of event a or the republicans, first of all, there's 2 ,200 republicans over 4 ,000 total in the table.

00:46 And does that question mark equal republicans given that they're 30 to 44? so the 30 to 44 total is 724.

00:58 And then the ones that are republican within that column there is 340.

01:06 So if you change those to decimals, maybe a little easier to see, we end up the first one is 0 .55.

01:14 That's the 2 ,200 divided by 4 ,000.

01:17 And the other one is 0 .47.

01:22 So those are not equal.

01:26 And because they're not equal, then the answer here is no.

01:30 They are not independent.

01:33 They are actually dependent.

01:34 So those are not equal.

01:36 So no, they're not independent.

01:39 For the next one, the probability of being a democrat is 1 ,800 over the total, 4 ,000...