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Problem 50 Hard Difficulty

Sampling senators The two-way table below describes the members of the U.S Senate in a recent year.
(a) Who are the individuals? What variables are being measured?
(b) If we select a U.S. senator at random, what’s the probability that we choose
a Democrat? a female? a female Democrat? a female or a Democrat?

Answer

a) VARIABLES=Gender (Male/Female) and Political affiliation (Democrats/Republicans)
b) $\begin{aligned} P(\text {Democrat}) &=\frac{60}{100} \approx 0.60 \\ P(\text {female})=& \frac{17}{100} \approx 0.17 \\ P(\text { female and Democrat }) &=\frac{13}{100} \approx 0.13 \\ P(\text { female or Democrat }) &=\frac{64}{100} \approx 0.64 \end{aligned}$

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Video Transcript

Okay, So for problem 50 were given this table here of Democrats and Republicans versus males and females and part Ai says, who are the individuals and what variables are being measured So we can see that are variables here? Released on the left part of the equation are the table inside is either damn a Republican in an up here. We're looking at male versus female. Okay, so that's that. Answers what variables we're looking at now Part B were asked to find several different probabilities, so were asked to find, uh, if we select any one of these at random, What's probably that we choose a Democrat? Probability of a female, a female Democrat, a female or a Democrat. So what? Start out with some basics here. First thing that I need to know how many total we have so ever add all these up. Just tell you what it is. It ends up equaling out. I'm gonna call end for our total is equal to 100. So me transferred to another page here. Um, if we're looking at the probability of a Democrat will just say d Well, let's go back to this page. We have if we're looking for the probability of a Democrat, we're looking at this row here, so let's add up everything in that row, we have 47 plus 13. 47 plus 13 equals 60 and we put that over. Our total 60 divided by 100 is 1000.6. That's our answer for that. Okay. Going back here, get rid of some of this stuff. Okay? So next one we're doing is we're looking at the probability of a female. So we're gonna be looking at this column here. All right, so we have 13 plus four, So the probability of female is 13 plus four. So 13 plus four of 17. And once again, we're doing that over our total. So we're going to get 0.17 because that does not would do this very well. Next thing we're going to be looking for is our is our probability, uh, being female and being democrat. Now, this one's a little bit more specific, so we have to not only look at the female side, but we also, so we have to look at female, but we also have to look at Democrat, so we're gonna go across the row and down the column. So the only one that intersects with both those is right here at 13. So going back over here, we do the same thing. We take our 13 and that goes over once again. 100 to give us 1000.13 for on answer. All right. And for the last one, we're looking at probably a female or a Democrat. So I'm gonna right this with a little union symbol, female or a Democrat. So the difference between the female and a Democrat and female or Democrat is that there's an intersection, whereas this one we can be either or so I'm gonna circle Democrat because you could be a Democrat or you could be a female. That means I'm adding up this row in this calm. But be careful not to double count that 13. Okay, so I'm gonna 47 plus 13 gives me my 60 and I have to add that four in down here now, 64. So we get 64 over 100 equals 1000.64 And there you have it, huh?