Two dice are thrown. The events A, B and C are as follows: A: getting an even number on the firs

Two dice are thrown. The events A, B and C are as follows:...

Key Concepts

Set Difference

Set difference is a concept used to describe an event that consists of outcomes belonging to one event while excluding those that are part of another. It is often colloquially referred to as 'but not', and it plays a key role when combining events to exclude certain outcomes, as seen in expressions like 'A but not C'.

Intersection of Events

The intersection of two or more events is the event that includes only those outcomes that are common to all the events involved. Representing the logical 'and' operation, intersections are essential when determining the likelihood of simultaneous or co-occurring events.

Union of Events

The union of two or more events refers to the event that includes any outcomes that belong to at least one of the individual events. This logical 'or' operation is used to combine events, representing the idea that an outcome can satisfy one condition, another, or both.

Complement of an Event

The complement of an event consists of all outcomes in the sample space that are not in the original event. It is commonly denoted by a prime symbol or a 'not' operator, and understanding complements is crucial for expressing events like 'not A' or 'not C' in probability problems.

Event

An event is a subset of the sample space that represents a collection of outcomes that share a certain property. Events can be simple, involving a single outcome, or compound, involving multiple outcomes, and they form the foundation for calculating probabilities in any random experiment.

Sample Space

In probability theory, the sample space is the complete set of all possible outcomes of a random experiment. It is the basis upon which events are defined, and understanding it is essential for determining the probabilities of various events, such as outcomes when rolling dice, drawing cards, or flipping coins.

Transcript

00:01 Has told us that we have two dice and they are thrown okay so what will be the sample space this will be the sample space these are all the possible combinations that can appear when two dice are thrown all right and there are three events that is given that a is an event which says that there is an event even even number on the first dice there is an event b which says that there is an odd number on first dice and there is an event c which says that when two dyes are thrown the sum of those two numbers will be less than or equal to five alright, now moving on to the solutions.

00:33 What does our question ask? we will solve it accordingly, all right? the first part that the question asks us is to find out a prime, all right? what is a prime? so instead of finding a prime, first what we'll do is that first we will find a, b, and c.

00:52 And then accordingly we'll find all the other things.

00:55 So let's see, let's start with a.

01:00 Okay, what will be a? let's see here.

01:02 It says a is the even number on first dice now which is the even number on first dice for a i think this will be the column of even number on the first dice right this one here then this one four the column of four and then the column of six okay this will be the event this will be the items here so here we will have two comma one two comma two all the way to two comma six here here we will have 4 .1, here we will have 4 .2 all the way to 4 .6.

01:36 And then here we will have 6 .1, here will have 6 .2 all the way to 6 .6.

01:42 Okay.

01:43 So this is a.

01:44 Now, what will be b? let's see what our b is? b says b is odd on the first dice.

01:53 So what is odd on the first dice here? this one.

01:57 Okay.

01:57 And then from 3 and then from 5.

02:01 Right so let's come down here and we'll have our events as 1 .1 1 .2 all the way to 1 .6 here we will have 3 .1 3 .2 all the way to 3 .6 and here we will have 5 .1 here we will have 5 .2 all the way 2 here we will have 5 .2 all the way 2 here we will have 5 .6 right and then we have the event c isn't it which was the sum of digits should be less than or equal to five so let's see for this we'll have less than five for this for this for this right now for two one two two and two three then three one three two you will have four one here and that's it so here we will have the items as one comma one one comma two here we will have 1 .3 here we will have 1 .4 all right then from here we will have 2 .1 2 .2 and then here we will have 2 .3 then here we will have 3 .1 and then 3 .2 and then here we will have 4 .1.

03:26 These are all the elements that will give us less than or equal to 5 right now the question has asked us okay we have already written solutions so we don't need to write this again here.

03:37 Now the question here has asked us to find a complement.

03:41 What does a complement mean? a complement means everything that is not a.

03:46 All right? so let's see what is not a.

03:51 Here is these three columns, isn't it? a here is these three columns, which is 2, 1 to 2 .6, 4 .1 to 4 .1 to 4 .6 and 6 .1 to 6.

04:07 So what is left? what is left is 1 to 1 .6.

04:11 3 .1 to 3 .6 and 5 .1 to 5 .6.

04:16 These are not in a, right? so our a complement here will be 1 .1, 1 .2 all the way to 1 .6.

04:28 All right.

04:29 Here we will have 3 .1, 3 .2 all the way to 3 .6.

04:34 Then you will have 5 .1 5 .2 all the way to 5 .6 now if you look at this carefully this is also equal to our b look at this this is also equal to our b right now let's move on to the second part the second part tells us to find what let's see the second part tells us to find not b okay not b so technically this writing this way this we can write this as as b complement this and not b means the same thing okay so what will be our not b let's see what is our b is this right 1 .1 to 1 .6 3 .1 to 3 .6 and 5 .1 to 5 .6 so except this what is left in the sample space here what is left here is 2 .1 to 2 .6 4 .1 to 4 .6 and then 6 .1 to 6 .6 right so that will be equal to our not b here.

05:34 So here we will have 2 .1 2 .2 all the way up to 2 .6.

05:40 Here we will have 4 .1.

05:42 Here we will have 4 .2 all the way up to 4 .6.

05:46 Here we will have 6 .1.

05:48 Here we will have 6 .2 all the way up to 6 .6.

05:53 Now if you look at this carefully, this is also equal to our a.

05:57 Right? here.

05:59 See this is a and this is our b0 .0 is equal to a.

06:02 Okay so now moving on to the third part what is our third part we need to find let's see we need to find a or b okay a or b means a union b which means everything in a and in b okay what is there in a and b add everything and make it one big sample space now what is our a is 2 .1 to this right you can see the a here this is our b now look at this carefully what is in a that is not in b and what is in b that is not in a but if you add a and b completely together what do we get we get our sample space that is equal to s here this is this is what we get if we do a union b okay so i'll not write the entire thing here you can do that okay i'll just write it here our a union b will be equal to sample space okay so sample space that is equal to s s is above there you can see that very easily all right now moving on to the fourth part what do we need to find a and b okay a and b this means a intersection b this means common between a and b this means what is there common in between a and b okay now let's go up and see this is a and this is b you can see there is nothing common okay all the elements in b and a are different thus a and b is equal to shy or null all right now moving on to the fifth part what do we need to find is a but not c okay a but not c so what does that mean a but not c this means a minus c okay everything in a but it should not not include parts from c.

08:18 All right.

08:19 So what was our a here? we had our a as 2 .1, 2 .2, up to 2 .6.

08:29 Here we had 4 .1, 4 .2 all the way up to 4 .6.

08:34 And then here we had 6 .1, 6 .1, 6 .2 all the way up to 6 .6.

08:41 Subtracting c what was our c 1 .1 2 .2 okay 1 comma 2 1 .3 and then 1 comma 4 here we will have 2 .1 alright and here we will have just a second here we will have 2 .3 then here we will have 3 .2 okay and then here we will have 3 .1 these are the elements in so if you look at this carefully what do we need to omit we need to omit 2 .1 this will cancel out 2 .2 will cancel out okay and also here 4 .1 4 .1 will also cancel out from here this will cancel out this will cancel out okay 2 .3 here we 2 .3 was present this will also cancel out right so ultimately what we are left with is 2 .4 2 .5 2 .6 here we will have 4 .2, there will have 4 .3 all the way up to 4 .6 here and then here are 6 .1, 6 .2.

09:54 This will remain the same because nothing was omitted from this line.

09:58 So this is equal to a but not c...

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