A department store sells sport shirts in three sizes (small, medium, and large), three patterns (plaid, print, and stripe), and two sleeve lengths (long and short). The accompanying tables give the proportions of shirts sold in the various category combinations.
(a) What is the probability that the next shirt sold is a medium, long-sleeved, print shirt?
(b) What is the probability that the next shirt sold is a medium print shirt?
(c) What is the probability that the next shirt sold is a short-sleeved shirt? A long-sleeved shirt?
(d) What is the probability that the size of the next shirt sold is medium? That the pattern of the next shirt sold is a print?
(e) Given that the shirt just sold was a short-sleeved plaid, what is the probability that its size was medium?
(f) Given that the shirt just sold was a medium plaid, what is the probability that it was short-sleeved? Long-sleved?
(a) .05 $\\$(b) .12$\\$ (c) .56, .44$\\$ (d) .49, .25$\\$ (e) .533$\\$ (f) .444, .556
All right, So we're given an inventory of shirts. Some of them were shortly. Some are long. See, some are small, medium large sum are plaid printer stripe. We're giving the distribution in this table here and were asked a bunch of probability questions about this inventory. But before we do that, I'm gonna do something real quick just to make my life a little easier in solving this problem. I'm gonna write in a total call him in a total row, both of these tables as such. Now let's start with the short sleeves and small rope. In order, find the total for this row. Ijust add everything up in the road. So four plus two of six six plus five is 11. So this is 0.11 and we'll continue going down the line. So let's see. This is eight and seven, which is 15 15 and 12. It's 27 and this is that's 10 18. Now what these distributions mean these numbers are the distribution of small, medium and large short sleeve shirts, respectively. I'm going to do the same thing down here with the columns now. So plaid, we have four plus 8 12 Totals 5 15 2799 and seven At 16 eight and 12 2025 25. We're also gonna fill in this bottom right hand corner here. We're just gonna add up either the total rose or the total column. It doesn't matter. Basically, this will give us the distribution of all shorts T shirts. All right, We're just gonna add these three because 15 and 25 is a nice 40. We can add 16 to that rather easily. That's 56. This should be the same as thes numbers Here. You can check that for years. I'm gonna do the same thing down here. So this adds up to 0.8 This adds up to 0.22 This adds up to 0.14 It adds up to 0.170 point 09 0.18 The shadow add up 0.44 Yes, and something we should also check real quick is these two numbers they add up to one. And this should make sense because the percentage of short sleeve shirts puts the percentage Long sleeve shirt should be ah, 100% of the shirts. Which is this one, all right. With that in mind, let's start solving these problems. Our first problem. We're supposed to find the probability of selecting a medium long print shirt. It's rather simple. We can just look in the table long sleeves, medium print and that 0.5 all right, probability of a medium print in general. Well, we just add the probability of finding medium print shortly, plus a medium print long sleep. So that will be here. I'm just going to color code them waas 0.5 should give us 0.12 So those are answers apart and be part see were given were asked for the probability of selecting a short sharp Well, we can't id did that already, actually with peace total columns. So there you go 0.56 Same here with long shirts 0.44 So there are answers. Alternatively, you could add up everything inside here, but like I've established, that's already the same of is what we did here. We just added less terms. All right. Number four are part D. I should say we need to find the probability of just selecting a medium certain. General. Here's where these total columns will come in handy. All right, let's look at short sleeve. The total distribution of medium shortly shirts 0.27 As for long seeds, 0.22 So this illegal 0.27 0.22 which is 0.49 All right. As for the prince will do the same thing. We'll look in the print for the short sleeves. 0.16 the prince for long sleeves 0.9 sarah 0.16 plus 0.9 equal 0.25 All right, here we have some conditional probabilities content with male. We need to find the probability of a selectee and medium short given that it was short and plaid. Now, your reminder that probability is defined as the number of successful outcomes over the number of total outcomes in this case are successful. Outcomes are the medium shirts within the subset and our total outcomes are all the short sleeved plaid shirts. So that would be this column of the table right here now. Of those the mediums All right here. So in our numerator will have 0.8 in our denominator will have 0.15 This equals 0.533 rounding to three decimal places finally part off. So we need to find the probability of finding a short sleeve shirt, given that it was a medium and plaid shirt. Once again, successes over over total outcomes. So our successes, our short sleeve medium plaid shirts, which are right here. So that goes in the numerous up? Yeah. No, that's great. So that goes in the numerator. My fault. I thought I was circling something else. That's our total welcomes. We would just add our medium plaid shorts T shirts with our medium plaid long sleeve shirts. So we're gonna throw that nominator. Uh, hurray for color coding. So this equals 0.8 over 0.18 This is equal to this Reduces down to eight for ninth. So that 0.4443 decimal places and what we could use a similar role. Similar methodology down here. Well, since we only have two conditions outside of the medium and plaid shorts lever long sihf, you know that the probability of getting a long sleeve shirt, giving them medium. Platt is one minus. The probability of not getting a long sleep start given medium and Platt or the probability of finding a short sleeve shirt giving that it's medium in plaid. And since we've already found that very conveniently, you could just subtract that out 0.5 by six and there you go.