WEBVTT Kind: captions Language: en 00:00:00.866 --> 00:00:04.826 >> Welcome to the Cypress College Math Review on Basic Probability. 00:00:07.366 --> 00:00:10.146 Objective one: Simple Probability. 00:00:11.116 --> 00:00:16.596 To find the probability of event E, P of E equals the number of ways 00:00:16.596 --> 00:00:21.756 that event E can occur, divided by the total number of outcomes in the sample space. 00:00:23.746 --> 00:00:30.866 Example one, in a pet store, there are 15 puppies, 22 kittens, and 18 rabbits. 00:00:32.036 --> 00:00:34.866 What is the probability of randomly selecting a rabbit? 00:00:35.356 --> 00:00:42.826 The probability of rabbit is equal to the number of rabbits, 18, 00:00:43.646 --> 00:00:46.406 divided by the total number of pets in the store. 00:00:48.466 --> 00:00:53.616 We add 15 plus 22, plus 18 to get 55 total pets. 00:00:54.156 --> 00:00:59.976 And as a decimal, this works out to 0.3273. 00:01:07.526 --> 00:01:08.336 Example two. 00:01:09.216 --> 00:01:13.666 The following data represents the number of classes a student is taking and their gender. 00:01:14.400 --> 00:01:19.180 What is the probability that a randomly selected student is taking one class? 00:01:21.840 --> 00:01:28.720 The probability of one class, is the number of students taking one class, 48, 00:01:29.336 --> 00:01:33.196 and that is the sum of 13 plus 35. 00:01:35.666 --> 00:01:38.086 Divided by the total number of students. 00:01:39.266 --> 00:01:44.696 This can be found in the bottom right portion of the table, and that's 247. 00:01:45.216 --> 00:01:49.976 And that division comes out to 0.1943. 00:01:55.636 --> 00:01:56.576 Example three. 00:01:57.226 --> 00:02:02.326 What is the probability of drawing a heart from a standard 52 card deck? 00:02:03.896 --> 00:02:07.496 Here is an illustration of the 52 cards in the deck. 00:02:09.106 --> 00:02:11.646 Notice that all the hearts are in the second row. 00:02:12.216 --> 00:02:18.206 The probability of drawing a heart is equal to the number of hearts, 00:02:19.146 --> 00:02:21.506 and you can count that there are 13 of them. 00:02:22.606 --> 00:02:25.926 Divided by the total number of cards, 52. 00:02:27.726 --> 00:02:31.906 Thirteen over 52 reduces to 1/4. 00:02:34.276 --> 00:02:36.996 Often when the numbers are small, we leave the answer 00:02:37.000 --> 00:02:40.520 in fraction form instead of converting it to a decimal. 00:02:44.080 --> 00:02:47.320 Now pause the video and try these problems. 00:02:55.380 --> 00:02:59.600 Objective two: Addition Rule for Disjoint Events. 00:03:00.706 --> 00:03:06.956 Definition: Two events E and F are disjoint, or mutually exclusive, 00:03:07.526 --> 00:03:09.686 if they have no outcomes in common. 00:03:10.276 --> 00:03:17.946 The Addition Rule for Disjoint or Mutually Exclusive Events says, "If two events E 00:03:17.946 --> 00:03:23.566 and F are disjoint, then the probability of event E or the probability 00:03:23.566 --> 00:03:33.346 of event F occurring is given by, P of E or F is equal to P of E plus P of F." 00:03:35.976 --> 00:03:36.886 Example four. 00:03:37.720 --> 00:03:44.120 What is the probability of drawing a king or a queen form a standard deck of 52 playing cards? 00:03:46.620 --> 00:03:51.760 Note that it is not possible for a card to be a queen and a king at the same time. 00:03:52.656 --> 00:03:58.786 Therefore, there are no events in common, and the events king and queen are disjoint events. 00:03:59.996 --> 00:04:02.316 So, we can use the rule stated above. 00:04:04.700 --> 00:04:11.620 Probability of king or queen is equal to probability of king, plus probability of queen. 00:04:13.320 --> 00:04:20.840 The probability of drawing a king is 4 over 52, because there are four kings in the deck. 00:04:22.276 --> 00:04:26.906 The probability of queen is the same, also 4 over 52. 00:04:28.376 --> 00:04:39.976 We add these to get 8 over 52, and this reduces to 2/13 and as a decimal, that is 0.1538. 00:04:44.546 --> 00:04:45.496 Example five. 00:04:46.316 --> 00:04:50.846 The following data represent the number of classes a student is taking and their gender. 00:04:51.900 --> 00:04:57.240 What is the probability that a randomly selected student is taking two classes or three classes? 00:04:59.160 --> 00:05:04.420 Notice that these two events are disjoint, because there are no outcomes in common. 00:05:05.136 --> 00:05:08.776 That is, there are no students who are taking both two classes 00:05:08.780 --> 00:05:10.840 and three classes at the same time. 00:05:12.040 --> 00:05:17.140 So, we can use our Addition Rule for Disjoint Events, which says, 00:05:17.256 --> 00:05:21.946 "The probability of two classes or three classes, is equal to the probability 00:05:21.946 --> 00:05:25.640 of two classes plus the probability of three classes. 00:05:26.960 --> 00:05:31.980 The probability of two classes is 64 divided by 247. 00:05:33.280 --> 00:05:37.980 The probability for three classes is 94 divided by 247. 00:05:39.380 --> 00:05:47.980 This adds up to 158 over 247, and the decimal approximation is 0.6397. 00:05:51.500 --> 00:05:53.980 Now pause the video and try these problems. 00:06:01.760 --> 00:06:07.560 Objective three: General Addition Rule: If two events are not disjoint, 00:06:07.886 --> 00:06:12.906 which means they do have outcomes in common, then you should use the General Addition Rule. 00:06:14.456 --> 00:06:21.216 Which says, "For any two events E and F, the probability of event E or the probability 00:06:21.216 --> 00:06:28.266 of event F occurring is given by, P or E or F equals P of E, 00:06:28.736 --> 00:06:35.536 plus P of F, minus P of E and F. Example six. 00:06:36.280 --> 00:06:42.680 What is the probability of drawing a diamond or a king from a standard deck of 52 playing cards? 00:06:44.880 --> 00:06:52.000 The probability of diamond of king is equal to the probability of diamond, plus the probability 00:06:52.000 --> 00:06:56.100 of king, minus the probability of diamond and king. 00:06:58.380 --> 00:07:06.040 Notice that there are 13 diamonds in the deck, there are four kings, and there is one card 00:07:06.216 --> 00:07:09.946 which is both a diamond and a king. 00:07:10.136 --> 00:07:18.746 Therefore, the probability of diamond is 13 over 52, the probability of king is four over 52, 00:07:19.766 --> 00:07:23.836 and the probability of diamond and king is one over 52. 00:07:26.480 --> 00:07:32.960 This works out to 16 over 52, which simplifies to 4 over 13. 00:07:34.340 --> 00:07:38.120 And as a decimal, that's 0.3077. 00:07:42.420 --> 00:07:43.320 Example seven. 00:07:43.966 --> 00:07:48.286 Here again is the table showing the number of classes a student is taking and their gender. 00:07:49.280 --> 00:07:54.640 What is the probability that a randomly selected student is a female or taking three classes? 00:07:55.920 --> 00:08:00.980 Notice that these two events are not disjoint, because they do have outcomes in common. 00:08:02.126 --> 00:08:06.326 Meaning, a student can be female and taking three classes at the same time. 00:08:07.666 --> 00:08:09.916 So, we must use the General Addition Rule. 00:08:11.116 --> 00:08:18.306 Probability of female or three classes equals probability of female plus probability 00:08:18.306 --> 00:08:23.716 of three classes, minus probability of female and three classes. 00:08:24.296 --> 00:08:34.116 From the table, we see 132 total females, so that probability is 132 over 247. 00:08:35.596 --> 00:08:42.086 There are 94 students taking three classes, so that probability is 94 over 247. 00:08:43.256 --> 00:08:47.706 And there are 56 students who are both female and taking three classes, 00:08:49.046 --> 00:08:52.656 so that probability is 56 over 247. 00:08:54.126 --> 00:09:02.486 When we work this out we get 170 over 247, which as a decimal is 0.6883. 00:09:04.266 --> 00:09:06.736 Now pause the video and try these problems. 00:09:15.246 --> 00:09:17.366 Objective four: Complement Rule. 00:09:18.426 --> 00:09:25.966 Definition: Let E be any event, then the complement of E denoted as E C is all 00:09:25.966 --> 00:09:32.236 of the outcomes in the sample space that are not outcomes in the event E. An easy way 00:09:32.236 --> 00:09:36.766 to understand the idea of complement is that it is the opposite of an event. 00:09:37.746 --> 00:09:42.876 For example, if the event is that it is going to rain today, then the complement would be 00:09:42.920 --> 00:09:44.940 that it is not going to rain today. 00:09:45.240 --> 00:09:52.556 The Complement Rule says that, "If E represents any event and E C represents the complement 00:09:52.556 --> 00:09:58.146 of E, then P of E C equals 1 minus P of E." 00:09:59.596 --> 00:10:09.496 And we can also say, P of E equals 1 minus P of E C. Example Eight, what is the probability 00:10:09.500 --> 00:10:14.440 of not drawing a spade card from a standard deck of 52 playing cards? 00:10:15.580 --> 00:10:21.180 Notice that the event not a spade is the complement of drawing a spade. 00:10:22.880 --> 00:10:27.400 So, P of not spade is equal to 1 minus P of spade. 00:10:28.460 --> 00:10:34.360 Because there are 13 spades in the deck, that probability is 13 over 52, 00:10:35.520 --> 00:10:39.380 the whole number one turns into 52 over 52. 00:10:40.040 --> 00:10:46.840 So, when you subtract the 13 you get 39 over 52, and that reduces to 3/4. 00:10:53.800 --> 00:10:54.780 Example nine. 00:10:55.466 --> 00:11:00.856 According to the Pew Research Group, 81% of teens use social media networks. 00:11:01.706 --> 00:11:06.976 What is the probability that a randomly selected teen does not use social media networks? 00:11:08.540 --> 00:11:12.740 Because of the word not, we know we are to use the Complement Rule. 00:11:14.640 --> 00:11:18.720 So, the probability of not social media user is equal 00:11:18.720 --> 00:11:22.380 to 1 minus the probability of social media user. 00:11:23.680 --> 00:11:28.740 In the problem, it tells us that 81% of teens use social media networks. 00:11:29.576 --> 00:11:34.006 This is a proportion, but we can also think of it as the probability 00:11:34.006 --> 00:11:37.736 that a randomly selected teen would use a social media network. 00:11:38.366 --> 00:11:44.166 So, we use the decimal form of the number .81 as this probability. 00:11:45.940 --> 00:11:51.100 Then computing 1 minus .81 gives is the answer, 0.19. 00:11:54.140 --> 00:11:56.720 Now pause the video and try these problems. 00:12:04.040 --> 00:12:08.240 Objective five: Multiplication Rule for Independent Events. 00:12:09.136 --> 00:12:14.016 Definition: "Two events E and F are independent if the occurrence 00:12:14.016 --> 00:12:20.416 of event E does not affect the probability of event F. Two events are dependent 00:12:20.726 --> 00:12:25.726 if the occurrence of event E does affect the probability of event F." 00:12:26.896 --> 00:12:33.156 The Multiplication Rule for Independent Events says, "If E and F are independent events, 00:12:33.626 --> 00:12:39.956 then the probability of event E and event F occurring is given by P of E 00:12:39.956 --> 00:12:47.166 and F equals P of E times P of F. Example 10. 00:12:48.576 --> 00:12:53.566 A die is rolled and a coin is flipped, what is the probability that the result 00:12:53.566 --> 00:12:57.336 of the die is a five, and the coin comes up heads? 00:12:59.706 --> 00:13:06.076 Here note that the outcome of the die does not effect in any way the result of the coin. 00:13:06.996 --> 00:13:09.236 Therefore, these events are independent. 00:13:09.696 --> 00:13:16.076 The probability of five and heads is equal to the probability 00:13:16.076 --> 00:13:19.006 of five times the probability of heads. 00:13:19.426 --> 00:13:26.026 When a die is rolled, there are six equally likely outcomes, and one of those is a five. 00:13:26.906 --> 00:13:30.226 Therefore, the probability of a five is 1/6. 00:13:30.826 --> 00:13:36.146 With a coin, there are two equally likely outcomes, heads and tails. 00:13:36.676 --> 00:13:39.786 So, the probability of getting heads is 1/2. 00:13:41.066 --> 00:13:43.486 We multiply to obtain 1/12. 00:13:48.156 --> 00:13:49.196 Example 11. 00:13:49.816 --> 00:13:54.596 According to the Pew Research Group, 95% of teens use the internet. 00:13:55.436 --> 00:13:58.066 Suppose four teens are randomly selected, 00:13:58.626 --> 00:14:01.806 what is the possibility that all four use the internet? 00:14:03.056 --> 00:14:07.706 Here we will use the proportion 95% as the probability 00:14:07.706 --> 00:14:10.806 that a randomly selected teen uses the internet. 00:14:11.496 --> 00:14:16.026 And we will use the decimal form .95 to represent this probability. 00:14:16.646 --> 00:14:23.876 Now the event that all four use the internet means that the first teen uses the internet, 00:14:24.246 --> 00:14:28.536 and the second uses the internet, and the third and the fourth. 00:14:29.576 --> 00:14:31.446 So, this is an AND probability. 00:14:32.746 --> 00:14:36.556 Furthermore, the four events are independent events, 00:14:37.196 --> 00:14:39.506 because whether one teen uses the internet 00:14:39.506 --> 00:14:42.780 or not does not affect whether the others use the internet. 00:14:44.100 --> 00:14:47.400 So, we can use the Multiplication Rule for Independent Events. 00:14:48.720 --> 00:14:53.636 Thus, the probability that all four use the internet equals the probability 00:14:53.636 --> 00:14:58.906 that the first uses the internet times the probability that the second uses the internet, 00:14:59.406 --> 00:15:03.866 times the probability that the third uses the internet, times the probability 00:15:03.866 --> 00:15:05.606 that the fourth uses the internet. 00:15:07.046 --> 00:15:10.696 Each one of these four probabilities is 0.95. 00:15:11.646 --> 00:15:15.976 We multiply them together, that can be written as 0.95 00:15:15.980 --> 00:15:21.220 to the fourth power, and that works out to 0.8145. 00:15:23.540 --> 00:15:27.760 Now pause the video and try these problems. 00:15:34.560 --> 00:15:37.360 Objective six: Conditional Probability. 00:15:38.246 --> 00:15:45.626 Definition: The Conditional Probability denoted as P of E given F is the probability 00:15:45.626 --> 00:15:50.026 that event E occurs, given that event F has already occurred. 00:15:51.046 --> 00:15:56.526 There are two ways to find Conditional Probability, if E and F are any events, 00:15:56.896 --> 00:16:05.476 then the Conditional Probability P of E given F can be found by, P of E given F equals P of E 00:16:05.476 --> 00:16:16.256 and F divided by P of F. Or, P of E given F equals the number of outcomes in E and F divided 00:16:16.256 --> 00:16:22.266 by the number of outcomes in F. Example 12. 00:16:22.966 --> 00:16:30.036 According to the U.S. National Center for health Statistics, in 1997 0.2% of deaths 00:16:30.036 --> 00:16:35.166 in the U.S. were of 25 to 34 year olds whose cause of death was cancer. 00:16:35.996 --> 00:16:42.116 In addition, 1.97% of all people who died were 25 to 34 years old. 00:16:43.526 --> 00:16:47.966 What is the probability that a randomly selected death is the result of cancer 00:16:48.340 --> 00:16:52.760 if the individual is known to have been 25 to 34 years old. 00:16:53.920 --> 00:17:00.540 So, we are finding P of death due to cancer, given 25 to 34 years old. 00:17:02.236 --> 00:17:10.226 So, death due to cancer is playing the role of event E, and event F is 25 to 34 years old. 00:17:10.806 --> 00:17:16.776 In the problem, it is giving us information about probabilities 00:17:16.816 --> 00:17:18.636 and not about number of outcomes. 00:17:19.626 --> 00:17:23.166 Therefore, we will use the formula on the left to compute our answer. 00:17:24.536 --> 00:17:29.626 So, we need to find the probability of cancer and 25 to 34 years old, 00:17:30.056 --> 00:17:34.146 divided by probability of 25 to 34 years old. 00:17:35.396 --> 00:17:43.276 The 0.2% number in the problem represents the proportion of deaths that were both of 25 00:17:43.276 --> 00:17:45.936 to 34 year old, and caused by cancer. 00:17:47.046 --> 00:17:51.246 So, this number written as a decimal will give us the probability 00:17:51.246 --> 00:17:54.496 for the numerator, and that's .002. 00:17:55.056 --> 00:18:03.206 The 1.97 percent in the problem is only people who were 25 to 34 years old. 00:18:04.036 --> 00:18:07.516 So, this number gives us the probability for the denominator. 00:18:08.746 --> 00:18:12.456 And we write that as a decimal as 0.0197. 00:18:13.706 --> 00:18:18.956 Dividing these two numbers gives the answer 0.1015. 00:18:23.396 --> 00:18:24.456 Example 13. 00:18:25.396 --> 00:18:28.606 Here again we use the data in the table showing the number 00:18:28.606 --> 00:18:31.036 of classes a student is taking and their gender. 00:18:32.436 --> 00:18:37.416 What is the probability that a randomly selected student is taking three classes given 00:18:37.420 --> 00:18:38.680 that the student is male? 00:18:40.440 --> 00:18:43.460 The key word here is given and that is what indicates 00:18:43.466 --> 00:18:45.556 that we are to use Conditional Probability. 00:18:47.056 --> 00:18:51.846 Also notice that the information in the problem doesn't tell us anything about probabilities, 00:18:52.326 --> 00:18:56.076 but it does give us the number of outcomes in various events. 00:18:57.400 --> 00:19:01.340 Therefore, we can use the second method for finding Conditional Probability. 00:19:03.260 --> 00:19:09.396 And we say, P of three classes given male is equal to the number of people 00:19:09.400 --> 00:19:14.920 who are taking three classes and who are male, divided by the number of people who are male. 00:19:16.160 --> 00:19:18.740 Both of these numbers we can find in the table. 00:19:19.766 --> 00:19:24.206 The number of people who are taking three classes and who are male is 38, 00:19:24.860 --> 00:19:31.680 and the number of people who are male is just the total of males which is 115. 00:19:32.880 --> 00:19:37.080 Notice that we do not use the total 247 in this problem, 00:19:37.916 --> 00:19:43.676 because when doing conditional probability, the denominator is limited to the event 00:19:43.676 --> 00:19:48.246 that is given, which in this case is the 115 males. 00:19:48.906 --> 00:19:54.686 That fraction works out to 0.3304. 00:19:57.066 --> 00:19:59.536 Now pause the video and try these problems. 00:20:06.286 --> 00:20:10.966 Objective seven: General Multiplication Rule for Dependent Events. 00:20:12.166 --> 00:20:16.756 If we have two events E and F, and event F does not affect event E, 00:20:17.476 --> 00:20:24.696 that is E and F are dependent events, then the probability that E and F both occur is given 00:20:24.696 --> 00:20:34.536 by P of E and F equals P of E times P of F given E. Example 14. 00:20:34.986 --> 00:20:39.726 suppose two cards are randomly selected from a standard deck of 52 playing cards. 00:20:40.586 --> 00:20:45.866 What is the probability that the first card is a club, and the second card is also a club 00:20:46.076 --> 00:20:48.216 if the draw is done without replacement. 00:20:49.496 --> 00:20:53.566 Without replacement means that after the first card is drawn it is set aside, 00:20:53.906 --> 00:20:57.156 and the second card is drawn from the remaining group of cards. 00:20:57.540 --> 00:21:02.436 The two events mentioned are dependent, because the probability 00:21:02.440 --> 00:21:07.440 that the second card is a club is affected by whether the first card is a club. 00:21:07.880 --> 00:21:13.540 If you draw a club first, then it makes it a little bit less likely you'll draw another club. 00:21:14.756 --> 00:21:18.816 If you don't draw a club with the first card, then it's a little bit more likely 00:21:18.900 --> 00:21:21.820 that you would draw a club on the second try. 00:21:23.600 --> 00:21:29.320 So, we will use the General Multiplication Rule to say that the probability of first club 00:21:29.326 --> 00:21:35.026 and second club equals probability of first club times probability 00:21:35.026 --> 00:21:37.816 of second club given first club. 00:21:38.316 --> 00:21:43.656 The probability of first club is 13 over 52, 00:21:44.276 --> 00:21:47.646 because there are 13 clubs in the deck out of the 52 cards. 00:21:49.166 --> 00:21:54.116 Now that first card is set aside, and so, there are only 51 cards remaining. 00:21:54.936 --> 00:22:00.436 That's the denominator for the next fraction is 51, and the numerator is 12 00:22:00.856 --> 00:22:05.006 because there are only 12 clubs remaining in the deck. 00:22:05.936 --> 00:22:14.796 When we multiply these fractions it's 156 over 2,652, which reduces to 1 over 17 00:22:15.486 --> 00:22:20.536 or you could write the answer as a decimal approximation 0.0588. 00:22:23.426 --> 00:22:25.906 Now pause the video and try these problems.