0:03 now we're going to talk about two basic
0:08 rules of probability the first one is
0:20 the probability of forget an event a or
0:24 event B can be written as the
0:29 probability of a and union with B this
0:31 is equal to the probability of event a
0:35 no current plus the probability of event
0:36 B occurring
0:39 minus the probability of event a and
0:47 event B occurring together so make sure
0:51 you know this now sometimes you may be
0:53 dealing with two events that are
0:56 mutually exclusive for mutually
1:00 exclusive events the probability of get
1:04 an event a and B together is zero so
1:07 therefore the probability of getting a
1:13 or B becomes P of a plus P of B if a and
1:18 B are mutually exclusive events but you
1:20 can always use this formula which will
1:22 always work regardless of what type of
1:25 events you're dealing with the next type
1:38 so let's say if well first let's talk
1:41 about conditional probability the
1:45 probability of a given B is the
1:50 probability of getting a and B divided
1:54 by the probability of getting B so what
1:56 this means is that if we multiply both
1:59 sides by P of B we can get this equation
2:02 the probability of getting a and B is
2:05 the product of the probability of
2:10 getting a given B times the probability
2:15 of event B occurring now what about this
2:18 scenario what is the probability of B
2:22 given a this is still the probability of
2:26 a and B occurring but divided by the
2:30 probability of a so if we rearrange this
2:33 equation by multiplying both sides by P
2:35 of a and we see that the probability of
2:40 a and B can also be written as the
2:44 probability of B given a time's the
2:48 probability of a so notice the
2:55 sometimes the two events that you're
2:58 dealing with might be independent events
3:01 an independent event is an event that
3:05 does not depend on another event for
3:08 independent events the probability of a
3:12 given B is equal to the probability of
3:15 event a occurring because a does not
3:18 depend on B likewise the probability of
3:22 B given a is equal to the probability of
3:25 event B occurring because B does not
3:30 depend on a so when dealing with
3:35 independent events we get this equation
3:41 the probability of a and B occurring
3:46 is simply the probability of a times not
3:49 plus but times the probability of event
3:53 B occurring if the events are
3:57 independent if not then you can get one
4:02 of these equations depending on what you
4:05 were given one thing I do want to
4:08 mention is that when you see a and B
4:10 occurring this means that these two
4:14 events are occurring at the same time it
4:17 doesn't mean a and then B which could be
4:21 different from B and then a so because
4:23 the events occur at the same time the
4:26 order is not relevant otherwise the
4:29 formulas may be affected so just keep
4:31 that in mind therefore the probability
4:35 of a and B occurring is the same as the
4:39 probability of B and a occurring because
4:41 these two events occur at the same time
4:44 so I just want to add that clarification
4:45 for those of you who may have questions
4:49 on it now let's go ahead and put this
4:53 information to good use Sarah is
4:56 deciding which courses she wants to take
4:59 in her next college semester the
5:02 probability that she enrolls in an
5:06 algebra course is 0.3 and the
5:08 probability that she enrolls in a
5:13 biology course is 0.7 t the probability
5:16 that she will enroll in an algebra
5:20 course given that she enrolls in a bio
5:25 course is 0.4 t so Part A what is the
5:28 probability that she will enroll in both
5:33 an algebra course and a biology course
5:36 so take a minute pause the video and
5:40 using the formulas that we discuss go
5:41 ahead and get the answers to these questions
5:43 questions
5:47 so first let's begin by writing down
5:50 what we know the probability that she
5:55 takes an algebra course that is P of a
5:59 is 0.3 zero and the probability that she
6:06 takes a bio course that's 0.7 zero the
6:12 bility of a given B that is that she
6:14 takes an algebra course given that she
6:18 takes a bio course that's point four
6:22 zero so with this information we can now
6:25 focus on Part A so we're looking for the
6:27 probability that she will take both
6:33 algebra and bio so what's the formula
6:37 for this this is equal to the
6:42 probability of a given B times the
6:48 probability of either B occur so we can
6:52 see that P of a given B is 0.4 0 and the
6:54 probability that she will take a bio
7:00 course is 0.7 0 times 7 is 28 so 0.4
7:04 times point 7 is point 28 so there's a
7:07 28% chance that she's gonna take both
7:12 algebra and biology now let's move on to
7:17 Part B what is the probability that she
7:21 will enroll in an algebra course or a
7:27 biology course so this time we're
7:32 looking for P of A or B so what's the
7:35 formula for this so this is going to be
7:37 based on the addition rule as opposed to
7:41 the multiplication rule so this is P of
7:50 a plus P of B minus P of a and B now the
7:53 probability that event a will occur is
7:57 0.3 and a probability that she's gonna
8:01 take the biology course is 0.7
8:03 the probability that she will take both
8:08 algebra and biology we have it here that
8:15 is 0.28 so let's go ahead and subtract
8:21 these numbers so 0.3 plus 0.7 minus 0.28
8:27 this is equal to 0.72 so there's a 72%
8:31 chance that she will take algebra or
8:35 biology now let's move on to Part C are
8:39 the two events independent what would
8:43 you say what are the requirements for
8:48 two events to be independent in order
8:51 for events a and B to be independent the
8:55 probability of a given B must be equal
8:59 to the probability of a are they equal
9:02 the probability of a given B we could
9:05 see that it's point four and a
9:08 probability of event a occurring is 0.3
9:12 so these two are not equal therefore the
9:17 two events are not independent of each other
9:23 now what about Part D are the two events
9:27 in mutually exclusive what would you say
9:29 in order for the two events to be
9:32 mutually exclusive the probability of
9:37 event a occurring and event B occurring
9:43 must be equal to zero is this true well
9:47 we know that a and B the probability of
9:49 these two events occurring is point two
9:51 eight because that's what we calculated
9:54 in Part A so it does not equal zero
9:58 which means that events a and B are not
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