Compound Probability of multiple events - Free Educational videos for Students in K-12

Compound Probability of multiple events - By tecmath

Transcript


00:00	Good day . Welcome to Tech Math channel . What
00:02	we're going to be able to look at in this
00:03	video is probability involving multiple steps or multiple events .
00:09	It's part of a series of videos where I've been
00:10	having a look at probability and it's gonna be a
00:13	few things we're going to cover in this particular video
00:15	. We're going to cover our independent independent events and
00:18	how they affect their calculations as well as the product
00:22	. In addition rules that come into play when we
00:24	look at multiple events occurring and probability . So I'll
00:28	to steal those . We get along . I think
00:29	it's probably the best way to get to the first
00:32	off . We're considering multiple events occurring and probability events
00:36	can be considered either being independent or dependent . Okay
00:42	, so what's the difference between these ? An independent
00:46	event is one where each event is not affected by
00:49	any other event . I'll give you an example of
00:51	this to say I was throwing a coin in the
00:54	air . I was tossing a coin . You can
00:56	imagine . Okay , so here's my coin and you
01:01	can imagine , you know , we have the first
01:04	toss and we have the second toss , we have
01:05	the third toss and so I was trying to get
01:07	heads on each one . But what you'd probably realise
01:09	is that each individual toss would not be affected by
01:13	what had previously happened . That is to say it
01:15	would be completely and utterly independent . So that's the
01:20	first type of event we can have . Okay ,
01:22	so you'd be fair to say that if you will
01:25	say looking individually at each one of these , you
01:27	would say no matter what particular toss . The problem
01:31	is say getting ahead would be one into no matter
01:34	which particular stage it was . So compare this to
01:37	a series of events with the probability of something occurring
01:40	is dependent on the previous event . The example I
01:43	give for this is say I had a bag of
01:47	marbles . Okay , here's my bag of marbles .
01:50	And in this bag of marbles I have two Red
01:54	Marbles and I have three blue marbles . So if
01:58	I was to ask you , okay , we're gonna
02:01	pick a marble out . What would be the probability
02:04	of getting a red marble ? And you look at
02:07	this and you say , okay , We have to
02:09	read marbles out of a possible total of five marbles
02:13	. Okay , that's that's all well and good .
02:15	What would be the probability of them picking out the
02:17	second red marble ? I'll just put that as to
02:20	what would be the probability of picking out a second
02:23	red marble . Given that you'd already picked out the
02:25	first red marble , Would it be two out of
02:29	5 ? And you might look at that and say
02:30	actually it's not going to be ? And this is
02:32	why if I was to pick up this marble ,
02:35	what you've noticed now is that we only have one
02:38	red marble out of a possible four Marbles . So
02:42	what's actually happened is the previous event has change the
02:47	probability of the next event . Okay , so ah
02:52	pretty much because this event depends on what has happened
02:55	in the previous event , it's said to be dependent
02:58	, so there's something to watch out for with these
03:00	occasionally . What happens is you will be told that
03:04	the marble gets picked out and then it gets replaced
03:07	. So I'll give an example this year , you
03:09	pick out this marble and we get rid of this
03:11	red marble . But then what happened ? And we
03:14	have that two out of five probability . But then
03:17	what happens is it gets put back in and this
03:19	is said to be with replacement . If this occurs
03:23	, you're going to notice that the probability is going
03:25	to be exactly the same because it's been put back
03:27	in . We now have is two out of five
03:30	. And that's the case . You treat this like
03:32	an independent event . Okay . Each event becomes independent
03:36	. I'm going to go through uh this particular example
03:39	and show you how you might work out a few
03:41	things with this . Okay , so let's have a
03:44	look at this example a little bit further to say
03:46	we had this bag and it has five marbles in
03:50	it . Two of them are red and three of
03:51	them are blue . We're going to pick out two
03:54	marbles , We're not going to do replacement . So
03:56	it's going to be a dependent probability we were dealing
03:59	about , we're going to work at the probability of
04:01	getting both read , we're going to work out and
04:03	probably getting both blue and we're gonna work at the
04:06	probability of getting one red and one blue , but
04:08	not necessarily in that order . Okay , so the
04:11	best way to do this and to show you is
04:14	we would first set up a tree diagram , so
04:16	we're gonna do that right now . So what we
04:20	initially can do is we could either pick out of
04:22	red or we could pick out a blue marble that's
04:26	in our very first picking out . And then what
04:28	would happen is we would have our second choosing of
04:32	marble . The first one would be we might have
04:34	got a read first and then we picked out a
04:39	red . Or we could have picked out of blue
04:40	or we could have been a blue first , we
04:42	could have picked out a red and we could have
04:44	got a blue for the second one . We're going
04:46	to work out our probabilities for each one of these
04:49	particular parts of this occurring . So let's just consider
04:54	this first part here to say we picked out a
04:56	red to start off with . So we pick up
04:59	this red here and what's the probability of that occurring
05:01	? You're going to see that the probability of that
05:03	occurring is as two out of five chance . What's
05:09	probably getting a blue ? There's a three out of
05:11	five chance . No problem so far . I guess
05:15	now , let's consider this particular pathway here . If
05:17	we want to take a red marble , in fact
05:20	, let's do that , let's get rid of one
05:21	of them . What would be the probability of getting
05:24	a red barbell for our second , picking and look
05:26	and say , okay , there's only one left .
05:29	So it's a one out of four chance . What
05:31	would be the probability of getting a blue marble ?
05:34	And you'd say , okay , there's three out of
05:36	four . All right for the next part say we
05:43	didn't pick a red marble out and said what we've
05:45	done is we've actually pick the blue marble out so
05:47	we'll get rid of one of those . What would
05:49	be the probability of getting and going down this path
05:52	? That would be the probability of getting a red
05:54	marble in your JK . That's two out of four
05:57	. What's it ? Probably getting a blue marble that's
06:00	also two out of four . So you got to
06:01	notice all these different probabilities , these dependent probabilities that
06:06	are different . So we can work out a few
06:09	things with this . Now say we're trying to work
06:11	out the probability of getting both red . So if
06:14	we follow the particular pathway you're going to notice that
06:17	we start with this pathway . The first red we
06:19	have a two in five chance of getting the second
06:23	red . We have a one in four chance of
06:25	getting . Yeah , this is the first rule .
06:28	This is a product rule , pretty much if we're
06:31	talking about something occurring in a series , this occurring
06:34	and then this occurring , we multiply to work out
06:37	the probability of that particular pathway occurring . Okay ,
06:41	so we've got a two and five chance of this
06:44	and a one in four chance of this . We
06:45	multiply these particular probabilities . Two times one is two
06:50	, Five times four is 20 . We have a
06:52	two and 20 chance . I could simplify that down
06:55	to one out of 10 . Okay , what's the
06:58	probability of getting both ? Blue ? Okay , we're
07:02	going to follow along this particular path where we've got
07:04	a blue blue , we got a three out of
07:07	five chance And then we have a two out of
07:10	four chance and once again we'd multiply these . Okay
07:15	, so 3/2 or three times two , sorry ,
07:18	is six , Five times four is 20 . And
07:23	this can be simplified to three out of 10 chance
07:27	. Okay , So that's the product rule . We
07:29	just if you have a something occurring in a series
07:32	like that , where it occurs and then this occurs
07:34	, okay ? And it's following along and you're trying
07:36	to work out the particular pathway , You multiply what
07:39	a bit a red and one blue . So we're
07:42	going to follow . I'll show you the two pathways
07:45	we cannot follow this pathway when we get a red
07:47	or blue , and we could follow this pathway where
07:50	we get a red or blue . Okay , So
07:52	let's have a look at those . In fact ,
07:54	I'm going to put those in different colours , Okay
07:57	? And we're going to draw them a little bit
07:58	different . So the first one is we have That's
08:03	two out of five chance and then we have a
08:05	threat of four chance . We're gonna multiply those for
08:10	the 2nd 1 . What we do is we have
08:13	a three or 4 kids And we have a two
08:19	out of four chance and we're gonna multiply those .
08:23	So what's what's the outcome when we do that ?
08:25	First off , two times 3 is six , five
08:31	times 4 is 20 . And this one we over
08:34	three times two is six and five times four is
08:37	20 . And all together , what's our total probability
08:41	here ? Now , you're going to say we're going
08:42	to end up doing the same thing , so we're
08:44	trying to work out the total probability of getting one
08:46	red and one blue , we can add these together
08:49	. Okay then six plus six is 12 and the
08:54	bottom number stays the same , so this is the
08:57	addition rule . Okay , so if you have basically
09:00	the same outcome occurring or you want to , you
09:03	know , it's basically this occurs and this occurs ,
09:05	you can add them . Okay , So that's our
09:08	two particular rules here . So let's go through a
09:11	few examples where you can work on these yourself .
09:13	So the steps I go through , if I was
09:16	doing these old workout first off , whether the events
09:19	were independent or dependent from one another , a major
09:23	up a tree diagram , I think I'd probably help
09:25	at the start and then I'd work out the probability
09:27	is using those product and addition rules . So let's
09:30	have a look at an example here . Okay ,
09:32	in this example , I'm gonna consider two students two
09:35	boys and one's name is Mick , that's Mick and
09:41	the other guy's name is Dave . All right .
09:45	Just draw Dave here kind of looks the same and
09:50	this is Dave . Now , what we're gonna be
09:52	saying for this , is that probably then getting a
09:55	question right in the test , It's slightly different probability
09:58	of getting it right is 0.6 the probably the day
10:02	of getting it right is 0.7 . So what we're
10:05	going to work out is the following . What is
10:08	the probability of both , correct ? Okay . Both
10:13	correct . What is the probability of they are correct
10:20	? And what's the probability of And what's the probability
10:24	of at least one crate ? At least one of
10:28	them getting it right ? So what you might want
10:33	to do when you answer this , I reckon you
10:35	should give it a go . I'd probably go through
10:37	a workout first . Is it Is each event independent
10:40	independent of one another . Drop your tree diagram .
10:43	Start working it out . Product Edition Rules . See
10:45	how you go . So he might have done this
10:49	already . Look , Oh , yes , because whether
10:53	or not he gets it right is not dependent on
10:56	whether or not Dave gets it right . It's not
10:58	dependent on Mick and whether Mick it's all right .
11:00	It's not dependent on day . These two events are
11:03	said to be independent . Okay , So we'll draw
11:06	a tree diagram for this . Okay ? So we've
11:09	got first off Mick and the chance of him getting
11:12	it right or getting it wrong and then we have
11:15	Dave and we could say about him getting it right
11:18	or wrong and him getting it right or wrong .
11:21	So what's the probability ? See I'd recommend go through
11:24	and put these in . Mc . The chance of
11:26	getting it corrected 0.6 and therefore the chance of getting
11:29	it wrong is 0.4 because total probability is going to
11:32	be one uh for Dave , he has a 0.7
11:36	chance of getting it correct . That's going to be
11:38	for both of those . And he's gonna have a
11:40	0.3 chance of getting it incorrect . So let's work
11:44	out our probabilities . So what's the probability of these
11:48	guys getting it both correct ? You're going to say
11:50	okay , it's going to be equal to we're going
11:51	to follow along this particular pathway here , Correent and
11:55	correct . 0.6 times 0.7 .6 times .7 is going
12:02	to be 0.42 . Okay , what's the chance that
12:06	both them getting it incorrect ? We're going to follow
12:09	along this pathway .4 times .3 0.4 ties 0.3 is
12:17	going to be equal to zero point 12 Okay ,
12:23	Okay , what's the chance of at least one of
12:25	them getting it correct ? There's a number of possibilities
12:28	where this occurs . We have this one . Okay
12:33	, we have this one where it goes along here
12:36	, we have this one , but we don't have
12:40	this one because they've got to correct on this one
12:42	. So let's work out the probabilities for these .
12:44	Uh First off , what we have is the chances
12:47	going along here , of both them getting incorrect .
12:49	We're gonna multiply this . 10.6 by 0.7 is 0.42
12:54	.6 times .3 , Which is going to be 0.18
13:00	. The chance of this particular scenario occurring with here
13:02	gets it wrong and he gets right is .4 times
13:04	.7 , which is 0.28 . And we're going to
13:08	add these together because these are all part of the
13:11	same particular outcome here . So .42 plus .18 is
13:17	going to be a .6 plus .28 is going to
13:21	be 0.88 . All right . How do you go
13:25	on that ? What about one more example ? Again
13:28	, here's the question . Out of a deck of
13:30	52 playing cards ? Two cards are chosen . What's
13:34	probably to get into reds ? What's the probably getting
13:36	to kings ? What's the probably getting in any ace
13:39	followed by any six ? And what's the probably getting
13:42	an ace of spades followed by any three . So
13:45	once again , what you probably might want to do
13:47	with these is you might want to go and think
13:49	are they independent or dependent events ? Probably not going
13:53	to draw a tree diagram for this one . I
13:55	reckon it would be pretty big . So I'd probably
13:57	just try to put down the probabilities as I was
14:00	going line of each event and then working out whether
14:03	I was going to do product or addition rules .
14:05	Okay , give it a fly anyway , see where
14:09	you went for the first one . What's the probability
14:11	of getting two reds ? Uh Well , the probability
14:14	of pulling the first went out , There's 26 red
14:18	cards in a deck out of 52 . And then
14:22	you take the first red card out and then you
14:24	will realize that there's only 51 cards left . And
14:27	because you've already got one of the reds , There's
14:30	only gonna be 25 of those guys left and you'd
14:32	have to multiply these through . That's a pretty big
14:35	number . 26 times 25 is going to be 650
14:40	52 times 51 is 2652 . This simplifies to 25
14:50	out of 102 chance . Or probability ? What about
14:55	the probability of getting to kings ? So the probability
14:58	getting one king to start off with ? Well ,
15:00	there's four kings , one of these shoes . So
15:03	there's four out of 52 chance . Then the probability
15:08	of getting a second king is well , there's now
15:11	a three left , but there's only 51 cards .
15:14	So we'd multiply these 34 times three is 12 .
15:17	Uh 52 times 51 is going to be 26 52
15:22	, which is equal to simplified one out of 221
15:29	. What's the problem in the beginning ? Any ice
15:32	followed by any 6 ? So probably getting any ice
15:37	you're going to look at and say , okay ,
15:39	they're probably getting any ice . It's four out of
15:41	52 , Multiplied by the probability of getting any six
15:46	with his 6th 4/6es in there . So it's four
15:49	of those Over 51 , which is going to be
15:52	four , out of uh 2652 . I'm gonna run
15:59	out of space on that one . But that that
16:00	actually sympathize to four out of 663 . What about
16:04	the probability of an ace of spades followed by any
16:08	three ? I think I'm gonna draw this down .
16:09	You know , I'm gonna run out of space otherwise
16:11	. Okay , so the ace of spades , there
16:13	is only one ace of spades , so that's one
16:15	out of 52 and any three . Well , there's
16:18	going to be four threes in their Heart of Diamond
16:21	, a club in a spade . And we're gonna
16:23	multiply that way , four by four out of 51
16:26	. And this will multiply to give 4/26 52 which
16:32	would simplify To give one out of 663 . Anyway
16:39	, How did you go with those ? So ,
16:40	it's just something to be aware of with those .
16:42	Just whether or not things are independent or dependent and
16:46	then how to use those products in addition rules .
16:49	I think on the next video , I'm just going
16:50	to give a few examples of these and just start
16:53	saying , okay , give a bit of a fly
16:54	what you've learned these last few videos and see how
16:57	you go anyway . If you liked this video ,
17:00	smash the like button , I think I smashed last
17:02	time . Absolutely destroy the like button , subscribe and
17:07	I don't want to plug it . But I will
17:08	, there is a patron of you actually want to
17:10	start saying what you want out there . That would
17:13	be really , really good . Uh , keep help
17:15	. Keep math free . Good at giving a plug
17:18	. Can't blame me by the merch . You either
17:20	you can also do that anyway . Hopefully this video
17:22	was some help for you . See you next time
17:25	. Bye .

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