Probability Tricks - By tecmath
Transcript
| 00:0-1 | Good day . Welcome to Tech Math channel . What | |
| 00:01 | we're gonna be having a look at in this video | |
| 00:03 | is some probability shortcuts used to solve some more complex | |
| 00:06 | probability questions . So sit back and enjoy and if | |
| 00:10 | you like the video , please remember to like and | |
| 00:12 | subscribe . So I'm just gonna launch straight into this | |
| 00:15 | . I'm going to just be going through a few | |
| 00:16 | different questions and get used to these . So let's | |
| 00:19 | go . So the very first question we have is | |
| 00:22 | this one , a bag contains seven balls . A | |
| 00:25 | bag contains four yellow and three blue balls . What | |
| 00:28 | is the probability of choosing to yellow and one blue | |
| 00:32 | ball out of that ? So we're gonna be choosing | |
| 00:33 | three balls . So here's our bag here . We're | |
| 00:35 | gonna try and choose to yellow and one blue and | |
| 00:37 | see what the probability that is . Now to do | |
| 00:40 | this , it's going to be really , really handy | |
| 00:42 | if we use combinations . So I'm going to do | |
| 00:46 | a quick recap of how to do basic combinations right | |
| 00:49 | now . This goes for about a minute so you | |
| 00:50 | might want to skip this over if you're already familiar | |
| 00:52 | with this . Okay quick recap on combinations , combinations | |
| 00:57 | were literally saying how many different ways we can select | |
| 00:59 | are different objects from a total of and came from | |
| 01:03 | a bigger group and it's this scary looking formula here | |
| 01:06 | which really is not very scary at all once you | |
| 01:09 | get into it . So how many different ways can | |
| 01:11 | we select two different objects from a total of five | |
| 01:15 | the way we work this out . Okay so we're | |
| 01:16 | selecting two different objects from a total of five . | |
| 01:20 | So we're gonna go uh five C . Two and | |
| 01:25 | we're going to follow this rule here . So n | |
| 01:27 | factorial literally is five factorial five times four times three | |
| 01:33 | times two times one . Okay And this is all | |
| 01:37 | going to go over and take away our factorial so | |
| 01:40 | and take away our five Take away two is three | |
| 01:43 | factorial three times two times one . And this is | |
| 01:47 | gonna be multiplied by r factorial . Which is two | |
| 01:49 | factorial two times one . They're going to see here | |
| 01:53 | . What happens is this council's out Okay . 32 | |
| 01:57 | or one here , Council out here . So really | |
| 01:59 | all we're left with is This particular part here ? | |
| 02:02 | This five times 4 over two times one . Now | |
| 02:07 | , I'm going to show you this shortcut how you | |
| 02:09 | can get away with not doing all of this literally | |
| 02:12 | when we're selecting two objects out of five . So | |
| 02:15 | we're gonna do this five factory but we're only going | |
| 02:17 | to come out this are times Okay ? So five | |
| 02:20 | times four . Okay , we're not going to do | |
| 02:22 | the whole bit , so it's only five factorial and | |
| 02:25 | coming out two times and then we're just going to | |
| 02:27 | have the two factorial on the bottom and we're gonna | |
| 02:29 | be using this for our probability question coming up . | |
| 02:32 | Okay so now you've got the basics of combinations there | |
| 02:35 | . Let's go through and solve this . Now . | |
| 02:37 | Just a little thing here you're going to notice that | |
| 02:39 | we have an end here . This quite literally means | |
| 02:42 | we are going to be multiplying as we go along | |
| 02:44 | . This is all what we're going to be doing | |
| 02:46 | is we're gonna be adding . Okay so this is | |
| 02:48 | a pretty much a common thing with probability . So | |
| 02:52 | how many different ways can we choose from four balls | |
| 02:56 | to yellow balls ? That's the way we write that | |
| 02:58 | . Yeah we're gonna be multiplying that by For the | |
| 03:02 | blue balls . How many different ways from three balls | |
| 03:05 | can we choose one of them ? And this is | |
| 03:08 | all going to go over ? Well the total number | |
| 03:11 | of Uh combinations that we could have for choosing three | |
| 03:16 | balls out of seven . And how much is that | |
| 03:18 | ? Well that's gonna be uh so seven C three | |
| 03:24 | . So what we can do now is we can | |
| 03:25 | go through and we can solve this . Okay so | |
| 03:27 | let's do that . Okay uh So what's this mean | |
| 03:31 | here ? This means as you just saw before four | |
| 03:34 | times three we're gonna go for factories we're only going | |
| 03:37 | to come out two times and this is going to | |
| 03:38 | go over two factorial two times one . This is | |
| 03:43 | going to be multiplied by the blue stuff here . | |
| 03:46 | The combinations of the blue ones were literally coming out | |
| 03:48 | three factorial but only one times which is three and | |
| 03:52 | this is going to go over one factorial . The | |
| 03:55 | total number of outcomes we have is well we have | |
| 03:59 | seven factorial but it's only coming at three times to | |
| 04:01 | seven times six times five and this is all going | |
| 04:05 | over three factorial three times two times one . Now | |
| 04:10 | we can go through and solve this , you can | |
| 04:12 | see we can work this out fairly simply here . | |
| 04:15 | Okay , so let's work out our little bits here | |
| 04:17 | we have Four times three , which is 12 divided | |
| 04:20 | by two , so 12 divided by two . This | |
| 04:23 | is equal to six Times 3/1 . Well this is | |
| 04:28 | just equal to three . Okay , so what do | |
| 04:31 | we have here ? Seven times six times 5/3 times | |
| 04:34 | two times one or three times to six . So | |
| 04:36 | this council's this part here And seven times 5 is | |
| 04:40 | 35 . So what do we have now ? six | |
| 04:44 | times through is 18 , This is over 35 and | |
| 04:49 | that's our answer . This is how you can go | |
| 04:50 | through and solve these types of questions really , really | |
| 04:53 | simply , what about I give you one to do | |
| 04:56 | right now ? Okay , so here's one for you | |
| 04:58 | to give you go to a bag contains four yellow | |
| 05:01 | and four blue balls all together . That's eight balls | |
| 05:04 | . What's the probability of choosing out of this to | |
| 05:07 | yellow and two blue balls ? So we're gonna be | |
| 05:09 | choosing four balls from this , so use your combinations | |
| 05:13 | and go through and do this . So first off | |
| 05:15 | for the yellow balls , how are we going to | |
| 05:17 | do this ? We had trying to choose out of | |
| 05:20 | four , two yellow balls , We're gonna be multiplying | |
| 05:25 | because we haven't and here , so this means we're | |
| 05:26 | gonna be multiplying , we're gonna be multiplying this boy | |
| 05:31 | for the blue balls here , we're trying to get | |
| 05:34 | out of for blue balls , we're trying to get | |
| 05:37 | two of them . This is all going to go | |
| 05:40 | over , you're gonna see here for the total number | |
| 05:43 | that we have added a total number of eight balls | |
| 05:48 | . We are trying to get four of them . | |
| 05:50 | So we're literally gonna go through and solve like this | |
| 05:54 | . Okay , so what does that equal ? Okay | |
| 05:57 | , let's have a look . Okay , so we | |
| 05:59 | have for factorial but certainly coming at two times , | |
| 06:02 | so four times three and that's going to go over | |
| 06:05 | two factorial , So two times one , this is | |
| 06:09 | going to be multiplied by uh Okay , same thing | |
| 06:13 | four times three over two times one . Okay , | |
| 06:18 | because it's the same sort of thing . Okay , | |
| 06:20 | and this is going to go over , Okay , | |
| 06:22 | we're going to come out eight fat Store . Um | |
| 06:24 | We're coming Out four times at eight times 7 times | |
| 06:28 | six times 5 . And this is going to go | |
| 06:32 | over four factorial four times three times two Times one | |
| 06:37 | . So we can solve this now . Okay , | |
| 06:40 | so how are we going to go do that ? | |
| 06:42 | So four times three is equal to 12 , divided | |
| 06:45 | by two is equal to six , so this is | |
| 06:48 | going to be equal to six and that's also going | |
| 06:51 | to be 66 times six . That's going to be | |
| 06:53 | on the top there . Okay , on the bottom | |
| 06:55 | here , we're gonna probably want to do a bit | |
| 06:57 | of counseling out . Ok . Two times three is | |
| 07:00 | equal to six . Okay , We can also council | |
| 07:04 | out here where we can go . Uh divided by | |
| 07:06 | four is two , and four divided by four is | |
| 07:10 | one . So We've got rid of all the bottom | |
| 07:12 | there , so two times seven times five is equal | |
| 07:16 | to 70 . Okay , what is this cycle ? | |
| 07:20 | six times 6 is 36 And this is over 70 | |
| 07:24 | . So , the probability of choosing to yellow and | |
| 07:27 | two blue balls out of this particular . Our scenario | |
| 07:30 | here is 36 out of 70 . So what do | |
| 07:33 | you think of that ? It's a nice easy trip | |
| 07:35 | , right ? Uh start getting these under your belt | |
| 07:37 | . They're not too bad . Now , I'm going | |
| 07:39 | to go to the next difference that we can have | |
| 07:41 | with this one . Okay , Okay , this is | |
| 07:42 | the next type of question . The bag contains five | |
| 07:45 | yellow and three blue balls , as you can see | |
| 07:47 | here . So all up we have a total of | |
| 07:49 | eight balls . And what's the probability of choosing to | |
| 07:53 | yellow or two blue balls ? And you can see | |
| 07:55 | the difference here , Is this word here or now | |
| 07:59 | ? Instead of multiplying , this means what we're going | |
| 08:01 | to be doing is adding . Okay , so let's | |
| 08:04 | go through and do this . Okay , uh it's | |
| 08:06 | the same sort of idea here . So for the | |
| 08:08 | yellow balls here , we have from five , we're | |
| 08:13 | gonna be choosing to uh we're gonna be then adding | |
| 08:16 | this to put the addition there . And we're gonna | |
| 08:19 | be adding this to uh for the blue balls here | |
| 08:22 | from three we're going to be choosing to all of | |
| 08:27 | this is then going to go over . Uh well | |
| 08:29 | that total outcomes here which is going to be uh | |
| 08:33 | so we're gonna be choosing to balls as you can | |
| 08:35 | see . Um so eight C . two . Now | |
| 08:40 | let's go through and work this out . All right | |
| 08:43 | , no problems here . Okay so first off I | |
| 08:45 | put a line that this is all going to go | |
| 08:47 | over and let's just work this out there . So | |
| 08:50 | this is literally gonna be five times 4/2 times one | |
| 08:55 | . Okay Because that's what's going to be right , | |
| 08:58 | this is gonna be an added to ah this part | |
| 09:01 | here . Now this one here we're literally coming out | |
| 09:05 | three times 2 times and this is going to go | |
| 09:08 | over to times one . All right . And on | |
| 09:12 | the base here you're gonna see here this eight C | |
| 09:14 | to uh to being chosen from eight here we're literally | |
| 09:18 | going to be coming from eight . We're gonna come | |
| 09:20 | out to seven and that's all because we're only coming | |
| 09:23 | at two times and this is going to go over | |
| 09:25 | two times one . The only difference here you're going | |
| 09:28 | to see is this addition here , round the multiplication | |
| 09:31 | so we can go through and solve that now . | |
| 09:32 | So what does this equal ? While the yellow part | |
| 09:35 | here ? The first part , 5/4 is equal to | |
| 09:37 | 20 divided by two is equal to 10 . This | |
| 09:42 | is going to be added to , well Three times | |
| 09:45 | two is 6 divided by two is 3 . All | |
| 09:49 | of this isn't going to go over eight times 7 | |
| 09:52 | . seven nights of 56 . Okay , divided by | |
| 09:56 | two is 28 . Okay so what does this equal | |
| 10:00 | ? Now ? Let's have a look . 10 plus | |
| 10:03 | three is 13 . 28 stays is 28 . So | |
| 10:07 | the probability of this occurring is 13/28 . So how | |
| 10:11 | did you go with that ? All right . What | |
| 10:12 | about to give you an example here ? Okay , | |
| 10:14 | this is your turn A bag contains four yellow and | |
| 10:17 | seven blue balls . That's 11 balls altogether . What's | |
| 10:20 | the probably choosing to yellow or ? Okay . It's | |
| 10:24 | in addition there to blue balls . So , we're | |
| 10:27 | gonna be choosing to balls . So what about you | |
| 10:29 | give this a go for the first one here . | |
| 10:31 | You can see that from four balls . Were trying | |
| 10:33 | to choose to to this we are going to be | |
| 10:36 | adding For the blue balls here . We're trying to | |
| 10:39 | choose from seven balls . Two balls . Okay , | |
| 10:44 | this is all going to go over . Okay . | |
| 10:47 | A total number of combinations here we are choosing from | |
| 10:51 | 11 balls were choosing to So let's work this out | |
| 10:54 | there . So this is equal to the following . | |
| 10:56 | Okay , so four C two is equal to four | |
| 11:00 | times 3/2 times one . Okay , and this is | |
| 11:05 | going to be added to because this is an awe | |
| 11:07 | to combinations for the blue balls here , we're going | |
| 11:10 | to be literally going seven times six because we're only | |
| 11:13 | coming at two times over two times one . All | |
| 11:17 | of this isn't going to go over 11 C two | |
| 11:20 | , which is going to be 11 times 10/2 times | |
| 11:26 | one . So let's go through and sold this now | |
| 11:28 | and see what this equals . So for the first | |
| 11:31 | part here , what we have is four times three | |
| 11:35 | which is 12 divided by two , this is equal | |
| 11:38 | to six , this is going to be added to | |
| 11:40 | seven . Sixes are 42 divided by two , is | |
| 11:45 | 21 So 21 . So this part here is going | |
| 11:48 | to be 11 times 10 , which is 100 and | |
| 11:51 | 10 divided by two , which is 55 . What | |
| 11:54 | does this all legal ? Well six plus 21 is | |
| 11:57 | equal to 27 And this is going to be over | |
| 12:01 | 55 . So that's the probability of choosing to yellow | |
| 12:06 | or two blue balls out of this particular bag here | |
| 12:10 | . So what do you think of that ? A | |
| 12:11 | nice handy little trick . I'll tell you what I'm | |
| 12:14 | going to give you one more question and this is | |
| 12:16 | gonna be a fair bit harder because what we're gonna | |
| 12:18 | do is now combined these ideas . Okay , for | |
| 12:21 | our last question here , we have a bag that | |
| 12:22 | contains eight balls and we're gonna be choosing four out | |
| 12:25 | of it who contains five yellow and three blue balls | |
| 12:29 | . And what we're saying is the probability of choosing | |
| 12:32 | either three yellow , one blue ball or choosing to | |
| 12:36 | yellow and two blue balls . Okay , so this | |
| 12:38 | is a bit more complex . Now , what we're | |
| 12:40 | gonna be doing is we're going to first off work | |
| 12:41 | out this part and we're going to work out this | |
| 12:44 | part Okay . And we're going to work out that | |
| 12:45 | total probability . Okay , so first off This particular | |
| 12:48 | party of the three yellow , 1 blue ball . | |
| 12:51 | Okay , so what is that for our yellow ? | |
| 12:53 | We are trying to choose from five , we are | |
| 12:56 | trying to choose three balls , we are going to | |
| 12:59 | be multiplying this by and this goes for our blue | |
| 13:03 | balls here , we've got three blue balls and we're | |
| 13:07 | trying to choose one out of it so that there | |
| 13:11 | Okay , this is going to be added to the | |
| 13:13 | following Okay , so we're going to be dealing now | |
| 13:16 | with this particular part here we are from five balls | |
| 13:20 | , See how many ways we can choose two of | |
| 13:22 | them . And to this we're going to be multiplying | |
| 13:24 | this by the number of different ways . We can | |
| 13:26 | choose these two balls here out of the three , | |
| 13:28 | so three C two , all of this is then | |
| 13:32 | going to go over Our particular total combinations here , | |
| 13:38 | which is going to be eight C four , wow | |
| 13:43 | , that's pretty nasty . Right , you can do | |
| 13:45 | it . Okay , so let's just work this out | |
| 13:47 | , we'll get at five C three , This is | |
| 13:48 | going to get changed to five times four times 3/3 | |
| 13:54 | times two times one . Okay , this is going | |
| 13:58 | to get multiplied by three C one , which is | |
| 14:01 | going to be equal to Ok , the number of | |
| 14:03 | different ways we could do that , which is just | |
| 14:05 | going to be equal to 3/1 to this . We're | |
| 14:09 | gonna be adding five C two . So we're literally | |
| 14:12 | gonna be coming out uh from five factorial two times | |
| 14:15 | or five times four and this is going to go | |
| 14:18 | over two times one . This is going to get | |
| 14:22 | multiplied by three C two . So literally what does | |
| 14:25 | this mean ? Uh Well it means that we're gonna | |
| 14:28 | be going three times two were coming at two times | |
| 14:31 | and we're gonna put this over two times one 8 | |
| 14:36 | C4 . This means we're choosing four from a total | |
| 14:39 | of eight here . It means that when we do | |
| 14:40 | this we are going to be getting the following okay | |
| 14:44 | we're gonna get right times seven times six times five | |
| 14:50 | and coming up four times and this is going to | |
| 14:53 | go over four . Factorial four times three times two | |
| 14:57 | times one , Wow . Okay so let's go through | |
| 15:00 | now and work this out . Like we're working out | |
| 15:02 | in situ okay , so five times four times three | |
| 15:06 | , this is equal to 60 divided by six . | |
| 15:10 | This is going to be equal to 10 . three | |
| 15:15 | divided by one . While this is equal to three | |
| 15:19 | . Now we're doing the right . Hopefully I've made | |
| 15:21 | a mistake yet . Five times four , which is | |
| 15:23 | 20 divided by two . This is equal to 10 | |
| 15:28 | man , I'm really hoping haven't made a mistake on | |
| 15:30 | this 13 times two is six divided by two is | |
| 15:33 | three . So I'll put a three down there . | |
| 15:36 | Okay , so what have we got ? 10 times | |
| 15:38 | three plus 10 times three . Ok , We should | |
| 15:41 | be able to do that . So let's work at | |
| 15:42 | this bottom part or can we do a bit of | |
| 15:44 | cancelling ? Yeah , four times two is equal to | |
| 15:47 | eight and three and six . Well , we could | |
| 15:52 | just multiply by two here instead . So Let's work | |
| 15:55 | this out now . Okay . 10 times three is | |
| 15:58 | 30 . 10 times three is 30 so 30 plus | |
| 16:01 | 30 is 60 . Okay . Seven times five is | |
| 16:05 | 35 times two . His 70 . Okay divided by | |
| 16:10 | one is just 70 there . So this is 60/70 | |
| 16:15 | . So wow . I hope I got the right | |
| 16:17 | answer there anyway . I think I did . I | |
| 16:18 | think I did but you're gonna see what a really | |
| 16:21 | really great treat that is . It's a bit of | |
| 16:23 | a mental gymnastics and that sort of deal . But | |
| 16:25 | once you get it sorted out there's absolutely no problems | |
| 16:27 | . Anyway , tell me how you went with that | |
| 16:29 | video . Tell me how you went with that particular | |
| 16:31 | trick . Hopefully with some help to you must say | |
| 16:33 | it was fairly diabolical trick to put down on video | |
| 16:36 | . So uh please like and please subscribe anyway . | |
| 16:41 | We'll see you next time . Bye . |
Summarizer
DESCRIPTION:
OVERVIEW:
Probability Tricks is a free educational video by tecmath.
This page not only allows students and teachers view Probability Tricks videos but also find engaging Sample Questions, Apps, Pins, Worksheets, Books related to the following topics.
GRADES:
STANDARDS:
