Experiment and Sample spaces

Almost everything in life can be considered as an experiment for example in the last article, we discussed there was this person who wanted to go to the supermarket so the experiment here is that the person goes to the mall and the outcome is whether he gets infected or not.

Similarly, in cricket, every delivery is an experiment and the outcome could be 0 to 6 runs(excluding special cases for the sake of discussion).

On the same lines, doing a blood test for a specific disease can be considered as an experiment and the outcome could be positive or negative.

And a student writing an exam is also an experiment and the outcome could be any one of the ‘n’ grades.

So, almost everything that we do can be defined as an experiment and we have certain outcomes(we could have one outcome in some cases/experiments) associated with that experiment.

In theory, a person could go to the supermarket infinite times and the outcome would be infected or not infected. And every time the outcome may change.

Mutually exclusive — if one of the outcomes occurs, then the other can not occur for example if, on a particular delivery, say the batsman scores 2 runs, that we can not have a score of 0 runs or 3 runs for that particular delivery. It could not be 2 runs and 3 runs and 4 runs simultaneously on the same delivery.

Collectively exhaustive — means that this is the complete set of outcomes as in the sample space, apart from that nothing is possible for example if someone writes an exam the possible outcome would be either of {A, B, C, D, E, F(fail)} but no other grade(essentially the total number of possible grades as defined by the Education board, no other grade is possible).

In a given experiment, only one of the outcomes from the universal set is possible.

The outcome in every trial is uncertain but the number of outcomes is certain.

So, every time the bowler bowls a delivery, it is uncertain whether the batsmen will score 1 runs or 2 runs or 3 runs and so on but it is certain that it is going to be 1 of the 7 possible values(0 to 6 runs, ignoring the special cases for the sake of discussion).

Let’s take some standard experiments to understand the concepts:

The simplest one is tossing one coin and when we toss a coin, again, in theory, we can repeat this process(tossing a coin) infinite times and hence it satisfies the definition of an experiment or a trial and it has a well-defined set of outcomes, it’ll either be a Heads or Tails.

We are representing the universal set using the omega.

Now, if we toss two coins simultaneously, then there a total of 4 possible outcomes, and these 4 outcomes are mutually exclusive(we are ignoring the case when the coin does not land on the surface).

Similarly, if we toss 3 coins simultaneously, there are a total of 8 possibilities

Let’s relate the number of outcomes to the concepts of counting:

Say we have a language which has only two alphabets ‘H’ and ‘T’ and we are trying to create a sequence in this language, when we are tossing 1 coin then the sequence is of length 1, when tossing 2 coins the sequence is of length 2 and similarly when tossing 3 coins, the sequence is of length 3. And any of the outcomes(‘H’, ‘T’) can be repeated in the sequence.

We can consider this as we have 2 elements in the original collection and we want to make a sequence of length 1 or length 2 or length 3 with repetitions and we know that the number of ways of doing this is given as: ‘nᵏ’

Let’s take another example:

When we roll a fair dice, then the number of outcomes is 6, when we simultaneously roll in 2 dice, we have a total of 6² = 36 outcomes:

So, we have a collection/possible outcomes of 6 elements{1, 2, 3, 4, 5, 6} and we have to make a sequence of length 2 such that the repetition is allowed, then the total number of possible outcomes is 6².

Let’s take another example:

Say we have a deck of cards, and we pick one card randomly, then the total possible outcomes, ways of picking one card would be 52.

Now let’s say we pick a card, we see the card and place back the card in the deck and again we pick a card(both the times we have the entire deck of 52 cards, the same card can be picked second time as well), so again this is a problem of creating a sequence of length 2 from a possible set of 52 values and the repetition is allowed, so the number of ways is going to be 52².

If the condition is such that the repetition is not allowed or say we’re supposed to pick 2 cards simultaneously from a deck of 52 cards, so in this case, we have 52 choices for the first card and 51 choices for the second card and the total number of ways of picking up two cards would be: ‘52 * 51'

All the examples that we discussed so far were involving discrete outcomes for example we had 52 outcomes in one case, 4 outcomes when the two coins are tossed simultaneously, they were not continuous outcomes. Here is one example, where we have continuous outcomes

Here is a square dartboard which goes from 0 to 1 and we are throwing darts on it, and assuming that the dart always lands on this board, that means all the possible outcomes lies in this ‘1 X 1’ square, in this case, we can define the possible outcomes in the universal set as any element (x, y) which is defined by the x, y co-ordinates such that both ‘x’ and ‘y’ are lying between 0 and 1.

This is a set containing infinite elements as these are the real numbers we are talking about.

Events of an Experiment

When we toss two coins simultaneously, these are the 4 outcomes which are possible and forms the universal set

An event is a set of outcomes of an experiment and the result set of an event will always be a subset of the sample space. Let’s say we are interested in the outcomes in which the first coin gives us heads

So, out of the total 4 outcomes(as in the sample space), there are two outcomes in the which the first coin results in heads, and in this is a subset of the sample space and this defines an event and that event being that the first toss/coin results in heads.

Similarly, we could have an event that both the coins results in Tails

Similarly, when we have the experiment involving a deck of cards, then the event that there are exactly 2 aces out of a total of 3 cards drawn from the deck is a subset of the sample space which contains all possible combinations of 3 cards and using the concepts of counting, we can even count the total number of elements in the sample space(two cards must be aces that is they must from a total of 4 aces and the remaining card could be anyone from the remaining 48 cards(excluding aces this time))

One thing to note is that the event itself is a set. And since it’s an event, we can use the operations on these sets.

Let’s take the example of rolling two dice and we are interested in two events, the first event is that the first dice shows up a 2 and the second event is that the second dice shows up a 4, and here is the list of possible outcomes for these two events

Now we can talk of an event that the first die shows up a 2 or the second one shows up a 4. As the word ‘or’ is used, we can be sure that we are talking about the union of two events/sets, so we want the element to be in the outcome of the first event or the second event

Similarly, we have the intersection of the events, say we are interested in the outcomes that the first dice shows up a ‘2’ and the second dice shows up a ‘4’, so we are looking for outcomes that lie in the outcome space of set A as well as set B

Similarly, we could have the complement of events, for example, the complement of the set ‘A’ would be that the first dice does not show up a 2, so of all the 36 possible outcomes, 6 lies in set A and the remaining would belong to complement of A

Multiple events:

Say we are dealing with a deck of cards and we have 3 possible events as depicted below and the union of the 3 would be that either the outcome belongs to set A or it belongs to set B or it belongs to set C

Now instead of the union, the intersection of the 3 events would be the shaded region in the below image, it will represent elements that are present in the all the 3 sets

Disjoint events

The simplest example would be the intersection of a set and its complement, we could have other events as well as disjoint events and not just the complement of an event

Similarly, we can define ‘n’ events as mutually disjoint

Here is an example say we are tossing 3 coins and say the outcomes of 3 events are as follows:

And hence these events are pairwise disjoint, the intersection of any 2 events from these 3 events is null.

Now in addition to this condition, if they also satisfy another condition that their union is equal to the sample space, then such events that they are mutually disjoint and if their union forms the sample space, then they are said to be partition the sample space.

Taking the same example of tossing two coins, we have the following:

References: PadhAI