WEBVTT 1 00:00:01.500 --> 00:00:02.333 Hello students. 2 00:00:02.333 --> 00:00:05.340 Here now we are moving forward to problem six. 3 00:00:05.340 --> 00:00:07.777 So I'm just gonna first read the problem. 4 00:00:07.777 --> 00:00:09.630 "A study is conducted to assess 5 00:00:09.630 --> 00:00:11.490 the impact of caffeine consumption, 6 00:00:11.490 --> 00:00:13.680 smoking, alcohol consumption 7 00:00:13.680 --> 00:00:17.220 and physical activity on cardiovascular disease. 8 00:00:17.220 --> 00:00:20.370 Suppose that 40% of participants 9 00:00:20.370 --> 00:00:22.620 consume caffeine and smoke 10 00:00:22.620 --> 00:00:25.410 if eight participants are evaluated, 11 00:00:25.410 --> 00:00:27.930 what is the probability that exactly 12 00:00:27.930 --> 00:00:31.047 half of them consume caffeine and smoke?" 13 00:00:32.040 --> 00:00:33.420 So here from the problem, 14 00:00:33.420 --> 00:00:34.440 we understand that 15 00:00:34.440 --> 00:00:37.320 we are using discrete data outcomes. 16 00:00:37.320 --> 00:00:39.240 So we are looking for participants 17 00:00:39.240 --> 00:00:41.490 who consume caffeine and smoke. 18 00:00:41.490 --> 00:00:43.860 So here we will have to use 19 00:00:43.860 --> 00:00:45.933 the binomial distribution formula. 20 00:00:47.850 --> 00:00:51.400 As always, I encourage my students 21 00:00:52.740 --> 00:00:54.963 to write down the formula first. 22 00:00:57.030 --> 00:01:01.080 Not only there is always partial credit involved 23 00:01:01.080 --> 00:01:04.740 with just even putting the formula correctly, 24 00:01:04.740 --> 00:01:07.503 but also it helps you understand the process better. 25 00:01:14.550 --> 00:01:17.620 (indistinct) eight successes equal to 26 00:01:19.170 --> 00:01:23.223 the formula N factorial, 27 00:01:24.420 --> 00:01:25.743 X factorial, 28 00:01:27.240 --> 00:01:29.783 X minus N factorial, oops. 29 00:01:36.360 --> 00:01:40.510 Multiplied by P to the power X 30 00:01:41.880 --> 00:01:45.060 multiplied by one minus P 31 00:01:45.060 --> 00:01:48.210 to the power N minus X. 32 00:01:48.210 --> 00:01:49.920 So the first thing I'd like to do here 33 00:01:49.920 --> 00:01:54.180 is kind of see what each of these represent for us. 34 00:01:54.180 --> 00:01:56.760 So X here is going to be four 35 00:01:56.760 --> 00:01:59.190 because our problem stated that, 36 00:01:59.190 --> 00:02:02.973 exactly half and exactly half of eight is four. 37 00:02:05.130 --> 00:02:08.010 Then (indistinct) N is going to be eight. 38 00:02:08.010 --> 00:02:10.470 Again, stated in our formula, sorry 39 00:02:10.470 --> 00:02:12.420 stating our problem statement, 40 00:02:12.420 --> 00:02:17.310 our probability here is 0.4 or 40%, 41 00:02:17.310 --> 00:02:21.960 and one minus P then would be of course 0.6 42 00:02:21.960 --> 00:02:25.350 because probability always adds up to one. 43 00:02:25.350 --> 00:02:29.820 So now I'm going to kind of place all of this 44 00:02:29.820 --> 00:02:31.690 and scoot over a little bit here 45 00:02:34.221 --> 00:02:37.290 and going to see what it's going to look like. 46 00:02:37.290 --> 00:02:38.370 So on the top here, 47 00:02:38.370 --> 00:02:41.380 we will have eight factorial 48 00:02:42.420 --> 00:02:44.610 divided by four factorial, 49 00:02:44.610 --> 00:02:46.623 eight minus four factorial. 50 00:02:47.640 --> 00:02:51.880 Then here it's going to be 0.4 51 00:03:00.090 --> 00:03:02.400 multiplied well not multiplied, 52 00:03:02.400 --> 00:03:04.740 it's to the power four 53 00:03:04.740 --> 00:03:06.820 and then it's going to be one minus 0.4 54 00:03:08.820 --> 00:03:10.980 then eight minus four. 55 00:03:10.980 --> 00:03:13.260 So this is basically what we have 56 00:03:13.260 --> 00:03:16.170 to now solve algebraically. 57 00:03:16.170 --> 00:03:17.130 So here again, 58 00:03:17.130 --> 00:03:22.130 I'm going to put all the numbers. 59 00:03:22.980 --> 00:03:24.420 I'm going to put it here. 60 00:03:24.420 --> 00:03:26.400 So basically what is a factorial? 61 00:03:26.400 --> 00:03:27.630 Factorial is just going 62 00:03:27.630 --> 00:03:29.700 from eight all the way to one 63 00:03:29.700 --> 00:03:31.470 and our book does provide it, 64 00:03:31.470 --> 00:03:33.000 but I will do this as well. 65 00:03:33.000 --> 00:03:35.250 So it's going to be eight times seven 66 00:03:35.250 --> 00:03:38.910 times six times five times four 67 00:03:38.910 --> 00:03:43.020 times three and times two and times one. 68 00:03:43.020 --> 00:03:46.200 And on the bottom it's going to be four factorial here. 69 00:03:46.200 --> 00:03:49.320 So it's going to be one times two 70 00:03:49.320 --> 00:03:52.020 times three times four 71 00:03:52.020 --> 00:03:54.600 and again eight minus four is again four. 72 00:03:54.600 --> 00:03:57.390 So it's gonna be another four factorial here 73 00:03:57.390 --> 00:04:00.450 and multiplication sign. 74 00:04:00.450 --> 00:04:04.500 So one times two times three times four. 75 00:04:04.500 --> 00:04:07.800 So now when you start solving this algebraically, 76 00:04:07.800 --> 00:04:09.660 you will see that this is quite simple 77 00:04:09.660 --> 00:04:12.240 as a lot of these will cancel each other out. 78 00:04:12.240 --> 00:04:16.290 So this will cancel this total part, okay? 79 00:04:16.290 --> 00:04:19.380 Now four times two we know is eight, 80 00:04:19.380 --> 00:04:22.110 So this will cancel this eight. 81 00:04:22.110 --> 00:04:23.550 Then we have three left. 82 00:04:23.550 --> 00:04:25.360 So we will multiply three 83 00:04:26.250 --> 00:04:31.250 and basically three times two is what? Six. 84 00:04:31.650 --> 00:04:36.650 So now we will have a lot of this kind of simplified. 85 00:04:36.930 --> 00:04:39.510 So now we have seven times two is 14 86 00:04:39.510 --> 00:04:41.550 and 14 times five is what? 70. 87 00:04:41.550 --> 00:04:43.623 So I'm just gonna write that here. 88 00:04:44.640 --> 00:04:48.900 Seven times two times five. 89 00:04:48.900 --> 00:04:50.520 And also from this side, 90 00:04:50.520 --> 00:04:53.133 we have 0.4, 91 00:04:54.953 --> 00:04:56.760 0.4 to the power four. 92 00:04:56.760 --> 00:05:00.817 That's going to be 0.026, 93 00:05:02.790 --> 00:05:06.923 and this is going to be 0.1296, okay? 94 00:05:08.160 --> 00:05:09.780 So again, put these down here. 95 00:05:09.780 --> 00:05:11.310 Again, I'm just writing them again 96 00:05:11.310 --> 00:05:13.410 just to make sure we are showing 97 00:05:13.410 --> 00:05:14.830 the entire total work 98 00:05:20.880 --> 00:05:23.280 and we do this multiplication. 99 00:05:23.280 --> 00:05:25.113 What we get is 0.235872, 100 00:05:32.760 --> 00:05:36.480 and that basically when we round it up 101 00:05:36.480 --> 00:05:39.400 and do the percentage, it is 23.6%. 102 00:05:42.090 --> 00:05:43.500 So basically what does that mean? 103 00:05:43.500 --> 00:05:45.120 That means going back 104 00:05:45.120 --> 00:05:46.987 to what our question asked us, 105 00:05:46.987 --> 00:05:50.160 "If eight participants are evaluated, 106 00:05:50.160 --> 00:05:53.160 what is the probability exactly half of them 107 00:05:53.160 --> 00:05:55.050 consume caffeine and smoke?" 108 00:05:55.050 --> 00:05:59.430 Well, that probability is 23.6%. 109 00:05:59.430 --> 00:06:01.140 Now, one thing I want to let you know here 110 00:06:01.140 --> 00:06:04.140 is the type of calculator you'll use. 111 00:06:04.140 --> 00:06:05.340 I know a lot of you 112 00:06:05.340 --> 00:06:08.070 may have those sophisticated scientific calculators 113 00:06:08.070 --> 00:06:09.870 and even though yes, 114 00:06:09.870 --> 00:06:11.370 they're fine, they're great 115 00:06:11.370 --> 00:06:12.570 but please keep in mind 116 00:06:12.570 --> 00:06:15.240 those calculators have these built-in memories 117 00:06:15.240 --> 00:06:17.820 and you have to kind of vacate the memories 118 00:06:17.820 --> 00:06:19.710 before you do new calculations. 119 00:06:19.710 --> 00:06:22.680 The programming is also a little bit more complex. 120 00:06:22.680 --> 00:06:24.390 So I suggest my students 121 00:06:24.390 --> 00:06:27.150 for this class just to use a simple calculator. 122 00:06:27.150 --> 00:06:29.610 Actually, a lot of the calculation that I just did here, 123 00:06:29.610 --> 00:06:30.873 I used my home, 124 00:06:32.280 --> 00:06:34.830 basically my mobile home phone, 125 00:06:34.830 --> 00:06:35.663 it's not my home phone 126 00:06:35.663 --> 00:06:36.900 it's my mobile phone, 127 00:06:36.900 --> 00:06:38.910 but my personal phone. 128 00:06:38.910 --> 00:06:41.940 And I use the calculator that is built-in in that phone. 129 00:06:41.940 --> 00:06:44.020 So that's kind of the simply 130 00:06:45.690 --> 00:06:48.120 approach you can take, okay? 131 00:06:48.120 --> 00:06:50.970 And the simpler the approach sometimes the better 132 00:06:50.970 --> 00:06:53.250 because we really are, 133 00:06:53.250 --> 00:06:56.010 kind of in an introductory level by statistics course. 134 00:06:56.010 --> 00:06:57.390 So there's no reason for us 135 00:06:57.390 --> 00:07:02.390 to use really any of those complex calculators, okay? 136 00:07:02.610 --> 00:07:04.500 So it's just my advice. 137 00:07:04.500 --> 00:07:06.180 Of course, if you know 138 00:07:06.180 --> 00:07:07.620 how to use those calculators, 139 00:07:07.620 --> 00:07:09.300 well, yes you can use them. 140 00:07:09.300 --> 00:07:10.560 A lot of those calculators 141 00:07:10.560 --> 00:07:13.290 will do a lot of the calculations for you. 142 00:07:13.290 --> 00:07:15.720 But you do still have to show me your work 143 00:07:15.720 --> 00:07:17.250 because I need to see that 144 00:07:17.250 --> 00:07:19.320 you are understanding the process. 145 00:07:19.320 --> 00:07:22.590 And if you really do not get the exact answer 146 00:07:22.590 --> 00:07:23.423 we are looking for it. 147 00:07:23.423 --> 00:07:24.660 You show me the process. 148 00:07:24.660 --> 00:07:27.990 You show me that you wrote down the correct formula. 149 00:07:27.990 --> 00:07:30.210 Of course there's going to be partial credit. 150 00:07:30.210 --> 00:07:32.670 So again, it is up to you 151 00:07:32.670 --> 00:07:34.200 what type of calculator you use 152 00:07:34.200 --> 00:07:35.523 but that's my suggestion. 153 00:07:37.380 --> 00:07:38.370 And that's about it. 154 00:07:38.370 --> 00:07:40.023 Now we will go to 6B.