Probability - addition and multiplication rules - By tecmath
Transcript
| 00:0-1 | Good day . Welcome to Attack Math Channel . What | |
| 00:01 | we're going to be having a look at this video | |
| 00:03 | is how to work out probability over multiple events . | |
| 00:07 | An example , this I say you maybe had a | |
| 00:09 | coin and you flipped it and you wanted to know | |
| 00:12 | how many different times you would get head . So | |
| 00:15 | you're going to flip it twice ? Did you ? | |
| 00:17 | Could you get heads twice ? Ok . So heads | |
| 00:19 | and then heads again . And what were the probability | |
| 00:21 | of that ? Or for maybe for example , what | |
| 00:23 | would be having a look at is where a bag | |
| 00:25 | of marbles like we have here had Yeah , three | |
| 00:29 | blue marbles and three red marbles . And what would | |
| 00:34 | be the probability if we were to draw out two | |
| 00:36 | marbles and maybe getting both of them being read . | |
| 00:39 | And so we're going to be looking at these types | |
| 00:41 | of questions . Okay , what you gonna notice with | |
| 00:44 | these these these probabilities occur of multiple events and are | |
| 00:47 | easy to work out . We just got to keep | |
| 00:49 | in mind a few things in this video , we're | |
| 00:51 | gonna be looking at some of these sort of things | |
| 00:53 | . So we're gonna be looking at product in addition | |
| 00:54 | rules , improbability as well as events , how they | |
| 00:57 | can be independent or dependent on each other and how | |
| 01:00 | these affect calculations . So let's just get on and | |
| 01:04 | have a look at these with a few examples and | |
| 01:06 | don't forget if you like this video , don't just | |
| 01:09 | sit there and lightly caressed the like button . Actually | |
| 01:11 | smash that like button , smash it . Hey if | |
| 01:14 | you haven't subscribed already please subscribe . Anyway , I'm | |
| 01:17 | just gonna look at a few examples . Okay for | |
| 01:19 | the first example we're going to have a look at | |
| 01:21 | this , we're going to be considering flipping a coin | |
| 01:24 | twice and just having a bit of a look about | |
| 01:26 | how we might work out various probabilities of outcomes that | |
| 01:29 | might occur . So to illustrate this , a really | |
| 01:32 | good way would be a tree diagram . So we | |
| 01:35 | have our first flip and we could get two possible | |
| 01:39 | outcomes , we get a head or a tail , | |
| 01:41 | so we get ahead or a tile . This would | |
| 01:45 | be our first flip . We have a second flip | |
| 01:48 | where we have also two possible outcomes . We get | |
| 01:51 | once again head or a tail or we could have | |
| 01:55 | got tails first and we could get a head or | |
| 01:59 | a tail . Now , just a couple of things | |
| 02:02 | , which is really important to probably get at this | |
| 02:04 | stage um which is this what you're going to notice | |
| 02:07 | ? The probability of each event ? Ikaria probability of | |
| 02:10 | getting ahead is one into the probability getting a tail | |
| 02:15 | is one and two . And so our first flip | |
| 02:17 | here probably getting ahead is one and two are probably | |
| 02:20 | getting a tale is one and two . We consider | |
| 02:22 | our second flip . You probably noticed really quickly that | |
| 02:26 | it's also the probability of getting ahead here is a | |
| 02:28 | one and two and the probability getting a tale here | |
| 02:30 | is one and two . That is to say that | |
| 02:32 | these particular events in the second flip are independent of | |
| 02:36 | the first . I'm actually going to write that down | |
| 02:38 | . It's a really important thing to get this word | |
| 02:40 | here , that this is in dependent . Okay , | |
| 02:45 | uh each particular uh , outcome or each probability is | |
| 02:51 | independent or each event is independent of the other event | |
| 02:54 | . Okay . It's not affected by it . So | |
| 02:57 | we have a half chance of this occurring and a | |
| 03:00 | half chance of this occurring . Okay , so you | |
| 03:03 | can see this so far , we've set this up | |
| 03:05 | and it's all pretty nice . So what is the | |
| 03:07 | probability of this happening ? What is the probability of | |
| 03:12 | getting ahead and ahead ? We can use our probabilities | |
| 03:17 | here to work this out . Okay , Because the | |
| 03:19 | probability getting ahead at the start is equal to a | |
| 03:23 | half and the probability of getting so this particular thing | |
| 03:27 | here , we can get ahead here and we can | |
| 03:29 | get a head here and there probably to get the | |
| 03:31 | second one is also a half . Now this gets | |
| 03:35 | to a first rule of probability with multiple events and | |
| 03:38 | that's the product rule , pretty much the probability of | |
| 03:42 | two or more events occurring together can be calculated . | |
| 03:44 | So two or more events getting ahead and getting ahead | |
| 03:47 | can be calculated simply by multiplying these individual probabilities . | |
| 03:51 | Okay , so the probability one times one is one | |
| 03:55 | , two times 2 is four . The probability of | |
| 03:57 | getting two heads Is one in 4 . Okay , | |
| 04:01 | what about the probability of getting uh Tales ? Tales | |
| 04:07 | and Tales , you probably look at this and go | |
| 04:09 | , okay , it's the same sort of thing that | |
| 04:10 | probably are getting their tails as a half . The | |
| 04:12 | probability of getting a tales is also half . And | |
| 04:16 | so we're talking about this event followed by this event | |
| 04:19 | , we're gonna multiply these , this is a one | |
| 04:22 | in four chance . Okay I'm just going to take | |
| 04:26 | this one step further and show you a different rule | |
| 04:28 | in this . Just give myself a bit of space | |
| 04:30 | here . So I'm going to get rid of these | |
| 04:32 | two probabilities here and I'm going to talk about a | |
| 04:35 | different thing that might occur . What about the probability | |
| 04:37 | of getting one head at one tail ? But not | |
| 04:42 | necessarily in that order it might be a tail and | |
| 04:45 | a head or head and the tail and you're gonna | |
| 04:47 | see what to do that . We actually have two | |
| 04:49 | different ways it can occur . We could first off | |
| 04:50 | get ahead and then a tail . Or we can | |
| 04:52 | get a tail and then ahead . So I'm gonna | |
| 04:54 | write both of these ones down . So first off | |
| 04:57 | , if we go head tail , the probability of | |
| 05:00 | getting that is half of getting that first head and | |
| 05:03 | then half , you might say okay we're gonna multiply | |
| 05:06 | those because they're in a , you know that particularly | |
| 05:08 | this event followed by this event , we're gonna multiply | |
| 05:11 | these , this is a one in four chance of | |
| 05:13 | getting ahead in a tail . The chance of getting | |
| 05:15 | a tail then ahead , it's also a half times | |
| 05:18 | a half . Okay , half 12.5 which is equal | |
| 05:23 | to a quarter . So this gets to our second | |
| 05:26 | rule when we're looking at multiple events and probability is | |
| 05:30 | this if we're talking about um two events that are | |
| 05:34 | mutually exclusive that are not affecting one . In other | |
| 05:36 | words , trying to find out the total probability . | |
| 05:38 | Save something like with the tail and a head here | |
| 05:40 | and there's a couple of different ways this can occur | |
| 05:43 | throughout outcomes . What we do is we're going to | |
| 05:46 | add these , we're going to add at quarter and | |
| 05:49 | a quarter to get the total probability because the head | |
| 05:52 | and the tail is the same as a tail and | |
| 05:54 | a head . So what's a quarter plus a quarter | |
| 05:56 | , you probably look at it and say , okay | |
| 05:58 | , that's two quarters . Okay , so the probability | |
| 06:01 | of that occurring , it's two quarters okay ? Or | |
| 06:03 | a half . So something to be aware of . | |
| 06:06 | Okay , so that's the product rule where we multiply | |
| 06:09 | these if we're looking at something occurring in a line | |
| 06:11 | like that and then we've got something which is occurring | |
| 06:13 | , you know , and we're saying we want this | |
| 06:16 | and this occurring , we're going to add them together | |
| 06:18 | . Okay , that's the addition rule . So I'm | |
| 06:21 | going to go through another example and show you just | |
| 06:23 | a variation of this . All right . In this | |
| 06:25 | example , what we're gonna have a look at is | |
| 06:27 | we have a bag and it has three blue marbles | |
| 06:30 | and two red marbles , and we're gonna take two | |
| 06:32 | marbles out . Now we're going to work out also | |
| 06:36 | . Now what are different probabilities of different outcomes could | |
| 06:39 | be in the probabilities of those outcomes occurring . So | |
| 06:42 | once again , let's draw up a tree diagram . | |
| 06:45 | So we have our first event where we're taking out | |
| 06:48 | the first marble . Okay , So the first marble | |
| 06:50 | you'll probably look at and say , okay , we | |
| 06:52 | could either end up with a blue marble or we | |
| 06:54 | could end up with a red marble . So we | |
| 06:58 | do that , we're gonna pull out the blue marble | |
| 07:01 | , we get rid of that for instance , and | |
| 07:02 | then the next event . What we can do is | |
| 07:04 | we might end up with a blue marble or a | |
| 07:08 | red marble or for the we might get a red | |
| 07:10 | one first . We could end up with a blue | |
| 07:13 | marble or a red marble . Okay so let's have | |
| 07:17 | a bit of a look here about the various probabilities | |
| 07:19 | of each individual party . Now this is something to | |
| 07:23 | be very very wary of because the probability of each | |
| 07:25 | event is not independent . Okay , each probability is | |
| 07:29 | each particular event is not independently other . I'll show | |
| 07:32 | you what I mean by that to say . For | |
| 07:34 | instance , I pull this first blue marble out . | |
| 07:37 | You can probably guess . Okay there's three blue marbles | |
| 07:39 | out of a total of five marbles . For the | |
| 07:42 | reds . There is two out of 5 to 5 | |
| 07:45 | chance of getting a red marble first . That's for | |
| 07:48 | our first removal for the second one . What you | |
| 07:51 | might notice is this if I was to remove , | |
| 07:54 | Okay , we pick a blue marble and we get | |
| 07:56 | rid of it . Now what's the actual probability getting | |
| 07:59 | a blue marble now ? Because these are not independent | |
| 08:03 | , they are dependent on one of this . Actually | |
| 08:04 | . Now depends the probability of this , depends on | |
| 08:07 | what happened here . We have two blue marbles now | |
| 08:10 | out of a possible for and to get a read | |
| 08:13 | . We have two out of four . Okay , | |
| 08:16 | maybe that didn't happen . And maybe what happened instead | |
| 08:19 | is we went down here and we picked the read | |
| 08:21 | out first , so we got rid of a red | |
| 08:22 | . What's the probability of getting a blue marble ? | |
| 08:25 | There's three out of four . The probability getting it | |
| 08:28 | . Red is one out of four . And this | |
| 08:30 | is an example of a dependent or write that down | |
| 08:34 | . These are where we have uh different events that | |
| 08:38 | are dependent on each other dependent , Okay . And | |
| 08:41 | it's something to be really really aware of because it | |
| 08:43 | changes these probabilities as we go . But if a | |
| 08:46 | hint here , what you might see is occasionally you'll | |
| 08:49 | see this described as two tables , marbles are taken | |
| 08:51 | out without replacement . If they say they are replaced | |
| 08:55 | , what we're talking about is the marbles get put | |
| 08:56 | back in , and what it would mean is that | |
| 08:59 | we would end up with independent type scenario . Okay | |
| 09:01 | . Where we'd end up with still five marbles in | |
| 09:04 | here , so it wouldn't really affect these later ones | |
| 09:06 | and it wouldn't affect the probabilities here . So you | |
| 09:09 | probably notice here that we can work out probabilities here | |
| 09:11 | . What's the probability of getting uh to blues or | |
| 09:17 | only go through each one of these ? What's the | |
| 09:19 | probability of getting a blue and a red ? What's | |
| 09:23 | the probability of getting a red and a blue ? | |
| 09:27 | Red and the blue ? What's the probability of getting | |
| 09:29 | a red and a red ? Okay , let's have | |
| 09:33 | a quick look at these and these are all going | |
| 09:35 | to end up being product ones . We're gonna multiply | |
| 09:37 | this week , gold on the product . You know | |
| 09:39 | , we've got a three and five chance of getting | |
| 09:41 | the first blue . We have a two in four | |
| 09:44 | chance of getting the second blue multiply these here . | |
| 09:48 | Two times three is equal to six . Five times | |
| 09:52 | four is equal to 20 . Now , I know | |
| 09:54 | you can simplify this further . I'm going to leave | |
| 09:56 | that to you . I'm not gonna do that right | |
| 09:57 | now because let's face it , I'm gonna run out | |
| 09:59 | of space if I do that . What about the | |
| 10:01 | probability of getting a blue than a red ? That | |
| 10:02 | is a 3-5 chance of getting the blue . And | |
| 10:06 | to get this red here , there's a two in | |
| 10:07 | four chance . So this also is a six and | |
| 10:10 | 20 probability this one here , we have a two | |
| 10:14 | in five chance of getting a red first and then | |
| 10:17 | a blue we have a three and four chance . | |
| 10:20 | I'm going to multiply those , we end up with | |
| 10:21 | a once again a six out of 20 probability to | |
| 10:25 | get to reds . There's a two and five chance | |
| 10:28 | at the start of actually getting the first red and | |
| 10:30 | then there's a one in four chance of getting that | |
| 10:33 | one second red . So two times one is 2/20 | |
| 10:37 | you're gonna notice that two plus six plus six plus | |
| 10:39 | six adds up to 20 . So all our probabilities | |
| 10:41 | are there and that's the product uh product all in | |
| 10:44 | action there . Now , I might also say , | |
| 10:46 | okay , we can actually do this a little bit | |
| 10:47 | differently and maybe I say , okay , what about | |
| 10:49 | a blue and a red , but not in any | |
| 10:51 | particular order ? You go , okay , well we'll | |
| 10:53 | have to add , This is six out of 20 | |
| 10:55 | , I can't even write this down here , I'm | |
| 10:56 | gonna be struggling for space anywhere , I'm gonna rub | |
| 10:59 | this out , I reckon I'll put it here . | |
| 11:01 | What's the probability of the one red at one blue | |
| 11:09 | ? You can sit there and go , okay , | |
| 11:10 | well we've got two ways . This can help , | |
| 11:12 | we can get a blue and a red , which | |
| 11:13 | is a six out of 20 , And we've got | |
| 11:16 | a red and blue , which is a six out | |
| 11:18 | of 20 , and we're going to add these , | |
| 11:20 | Okay , This is a 12 out of 20 probability | |
| 11:23 | of occurring . Okay , so this is an example | |
| 11:29 | of a dependent event . Okay , Where an event | |
| 11:32 | actually changes the probabilities of each little part here and | |
| 11:36 | something to be wary of . What about one more | |
| 11:38 | example ? Okay , I'm gonna not vary this one | |
| 11:41 | up too much because I think it maybe it's a | |
| 11:43 | good idea to this at this stage , we're gonna | |
| 11:44 | go to marbles taken out , same sort of bag | |
| 11:47 | , we have two reds and three blues , but | |
| 11:48 | this time with replacement . Let's see what happens . | |
| 11:51 | Okay , so if we take the blue marble first | |
| 11:54 | , you have a three and five chance to get | |
| 11:57 | a red marble in the first drawing , you have | |
| 11:59 | a two and five chance . Now imagine we are | |
| 12:03 | , take this blue marble out . Here we go | |
| 12:06 | , we're gonna take it out but it's gonna get | |
| 12:07 | replaced . Okay , so that means we're putting it | |
| 12:10 | back in , so I've taken it out , but | |
| 12:11 | now I'm putting it straight back in . So what's | |
| 12:13 | the chance now of getting a blue marble ? And | |
| 12:16 | you go , okay , well it's still actually three | |
| 12:17 | out of five And to get a red is two | |
| 12:21 | out of five . Okay , you're going to notice | |
| 12:23 | that the actual probabilities are not changing here . And | |
| 12:26 | if we almost treat this event , were we not | |
| 12:28 | almost we exactly treat this event as independent because these | |
| 12:32 | outcomes here are not affected . These events here are | |
| 12:35 | not affected by this previous event . This would be | |
| 12:37 | a three out of five and this would be it's | |
| 12:39 | two out of five . And so hence our overall | |
| 12:42 | probabilities would change . Remember this probability of getting now | |
| 12:45 | say blue and blue is we're gonna multiply these through | |
| 12:50 | , it's gonna be three out of five times three | |
| 12:53 | out of five , which is going to be three | |
| 12:55 | times three is 9/25 . We're gonna probability getting say | |
| 13:00 | a blue and a red , blue and a red | |
| 13:03 | is going to be equal to three out of five | |
| 13:06 | , Talks to 85 , which is going to be | |
| 13:09 | six out of 25 . Okay . Notice the probabilities | |
| 13:12 | are all of a sudden different . The probabilities of | |
| 13:15 | getting a red and a blue is equal to two | |
| 13:20 | out of five times three out of 52 out of | |
| 13:21 | five times three out of five , Which is going | |
| 13:24 | to be equal to six out of 25 . The | |
| 13:26 | probability of getting a red and a red , it's | |
| 13:30 | equal to two out of five Times two out of | |
| 13:35 | five , Which is going to be four out of | |
| 13:38 | 25 . So be really , really careful of these | |
| 13:41 | when you do these , that if its replacement , | |
| 13:44 | that you are going to treat it differently too , | |
| 13:46 | It's not replaced . Okay . Uh so , you | |
| 13:50 | know , you can actually now say , okay , | |
| 13:51 | what's the probability , You can always say ? What's | |
| 13:53 | the probability of two blues or to read this may | |
| 13:55 | be the probability of um at least one blue . | |
| 14:05 | What's the probability of this ? Well this one has | |
| 14:08 | at least one blew this one has at least one | |
| 14:10 | blew . This one has at least one blue . | |
| 14:12 | It's out of 25 because we're gonna add all these | |
| 14:16 | together . So 25 25 25 the dog when they | |
| 14:19 | stay in the same nine plus six plus six is | |
| 14:22 | 12 . We have 21 it's 21 at 25 probability | |
| 14:28 | . Tell you what we'll do one more . Okay | |
| 14:30 | , in this example what we're gonna have a look | |
| 14:31 | at is a scenario . We have six apples where | |
| 14:34 | three of them are good and three of them are | |
| 14:36 | bad . We're going to take two out at random | |
| 14:38 | . Okay ? So let's draw up a tree here | |
| 14:41 | we have Good bad . That's our first one . | |
| 14:44 | We have Good , bad , good , bad . | |
| 14:47 | I reckon you should give this a go without waiting | |
| 14:49 | for me by the way , we're gonna take these | |
| 14:51 | out at random and remember we're not actually putting them | |
| 14:54 | back , there is no replacement . So what I | |
| 14:55 | recommend you go through first . Can you work at | |
| 14:57 | the probabilities of each of these particular outcomes of getting | |
| 15:01 | a probability of getting a good good , the probability | |
| 15:04 | of getting a good bad , the probability of getting | |
| 15:08 | a bad good and the probability of getting a bad | |
| 15:12 | , bad and I will sleep that one . We'll | |
| 15:14 | see how we go with those ones , go for | |
| 15:17 | it . So first off go through and work out | |
| 15:20 | your probabilities of each particular event and then go from | |
| 15:23 | there using those product rules . Okay , So give | |
| 15:26 | me the floor . Hopefully you did . All right | |
| 15:29 | , okay . The probability of getting a good apple | |
| 15:31 | to start off with is three and a six or | |
| 15:33 | a half . Its three out of six here as | |
| 15:35 | well . Um If you choose a good apple first | |
| 15:39 | , I'm just gonna so we choose one of these | |
| 15:43 | , we'll get rid of it . So what's our | |
| 15:46 | probability now of getting a good apple you might say | |
| 15:49 | ? Okay , it's two out of five . Probably | |
| 15:52 | getting a bad apple is three out of five because | |
| 15:54 | these are actually a dependent , particular dependent events here | |
| 16:00 | . Okay , well maybe that didn't happen , Maybe | |
| 16:02 | my apple was okay . And there it is , | |
| 16:06 | and instead what we did is we took a bad | |
| 16:08 | apple at first . Okay , let's have a look | |
| 16:10 | at what happens here . You know , the probability | |
| 16:12 | of getting a good apple . Is there going to | |
| 16:14 | be three out of five ? We have three good | |
| 16:17 | apples out of five apples . The probably are getting | |
| 16:19 | a bad apple is going to be two out of | |
| 16:21 | five . Okay , what's our different probabilities here ? | |
| 16:24 | Probably getting uh two good apples is three out of | |
| 16:28 | six Times two out of five . We're just following | |
| 16:33 | up their pathway there . 3 to 665 to 30 | |
| 16:37 | . What's probably getting a good and a bad , | |
| 16:39 | you might go , okay , that's a three out | |
| 16:41 | of six Times three out of five , which is | |
| 16:45 | going to be three , threes and nine out of | |
| 16:48 | 30 . The probability getting a bad and a good | |
| 16:53 | we have three out of six times three and 53 | |
| 16:57 | out of six times three out of five , Which | |
| 17:00 | is gonna be nine out of 30 or so , | |
| 17:03 | and they're probably getting two bads is three out of | |
| 17:05 | six times two out of 53 out of six times | |
| 17:08 | two out of five , Which is going to be | |
| 17:11 | six out of 30 . There you go . How'd | |
| 17:15 | you go that now ? If I was to say | |
| 17:16 | once again , what's the Probably getting a good apple | |
| 17:19 | and a bad apple ? You probably go cable . | |
| 17:21 | And I might say now , what's the probability of | |
| 17:23 | getting a good apple and a bad apple ? And | |
| 17:26 | you might say okay in any order we could add | |
| 17:29 | these two together . Nine out of 30 plus nine | |
| 17:31 | out of 30 would give us at least getting a | |
| 17:33 | good apple and are getting a bad apple . Would | |
| 17:35 | give us an 18 out of 30 probability anyway , | |
| 17:40 | look , hopefully you found this is some use that | |
| 17:42 | are the Product Edition rules , multiple probability events there | |
| 17:46 | . Hopefully this video is some help to you . | |
| 17:50 | If it was please like it , please subscribe . | |
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| 18:03 | free mouth I can get out there . Look , | |
| 18:04 | I'm going to keep making anyway , but your support | |
| 18:06 | would be lovely anyway . You're also gonna see , | |
| 18:08 | I've been putting up feeds , um , asking what | |
| 18:11 | videos do you guys want made and you might have | |
| 18:14 | a particular preference and if you do put them up | |
| 18:17 | there and I'll look at that are making them in | |
| 18:18 | the future anyway . Thanks for watching . See you | |
| 18:21 | next time . |
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