RANDOM EXPERIMENT: CONCEPT, SAMPLE SPACE, EXAMPLES - DUDAS - 2025

We speak of a random experiment when the result of each particular trial is unpredictable, even though the probability of occurrence of a certain result can be established.

However, it should be clarified that it is not possible to reproduce the same result of a random system with the same parameters and initial conditions in each trial of the experiment.

Figure 1. The roll of the dice is a random experiment. Source: Pixabay.

A good example of a random experiment is the rolling of a die. Even if care is taken to roll the die in the same way, each attempt will yield an unpredictable result. Actually, the only thing that can be said is that the result can be one of the following: 1, 2, 3, 4, 5 or 6.

Tossing a coin is another example of a random experiment with only two possible outcomes: heads or tails. Although the coin is thrown from the same height and in the same way, the chance factor will always be present, resulting in uncertainty with each new attempt.

The opposite of a random experiment is a deterministic experiment. For example, it is known that every time water is boiled at sea level the boiling temperature is 100ºC. But it never happens that, keeping the same conditions, the result is sometimes 90 ºC, other 12 0ºC and sometimes 100 ºC.

Sample space

The set of all possible outcomes of a random experiment is called the sample space. In the random experiment of rolling a die, the sample space is:

D = {1, 2, 3, 4, 5, 6}.

On the other hand, in the toss of a coin the sample space is:

M = {heads, tails}.

Event or occurrence

In a random experiment, an event is the occurrence or not of a certain outcome. For example, in the case of a coin flip, an event or occurrence is that it comes up heads.

Another event in a random experiment could be the following: that a number less than or equal to three is rolled on a die.

In case the event occurs, then the set of possible results is the set:

E = {1, 2, 3}

In turn, this is a subset of the sample space or set:

M = {1, 2, 3, 4, 5, 6}.

Examples

Below are some examples that illustrate the above:

Example 1

Suppose two coins are tossed, one after the other. It asks:

a) Indicate if it is a random experiment or, on the contrary, a deterministic experiment.

b) What is the sample space S of this experiment?

c) Indicate the set of event A, corresponding to the result of the experiment being heads and tails.

d) Calculate the probability that event A occurs.

e) Finally, find the probability that event B occurs: no heads appear in the result.

Solution

A bag contains 10 white marbles and 10 black marbles. Three marbles consecutively are drawn from the bag at random and without looking inside.

a) Determine the sample space for this random experiment.

b) Determine the set of results corresponding to event A, which consists in having two black marbles after the experiment.

c) Event B is to obtain at least two black marbles, determine the set B of results for this event.

d) What is the probability that event A occurs?

e) Find the probability that event B occurs.

f) Determine the probability that the result of the random experiment is that you have at least one black marble. This event will be called C.

Figure 2. Black and white marbles for random experiments. Source: Needpix.

Solution to

To construct the sample space, it is useful to make a tree diagram, like the one shown in Figure 3:

Figure 3. Tree diagram for example 2. Prepared by Fanny Zapata.

The set Ω of possible results of extracting three marbles from a bag with the same number of black and white marbles, is precisely the sample space of this random experiment.

Ω = {(b, b, b), (b, b, n), (b, n, b), (b, n, n), (n, b, b), (n, b, n), (n, n, b), (n, n, n)}

Solution b

The set of possible outcomes corresponding to event A, which consists of having two black marbles is:

A = {(b, n, n), (n, b, n), (n, n, b)}

Solution c

Event B is defined as: "having at least two black marbles after having randomly drawn three of them." The set of possible outcomes for event B is:

B = {(b, n, n), (n, b, n), (n, n, b), (n, n, n)}

Solution d

The probability of having event A is the quotient between the number of possible outcomes for this event, and the total number of possible outcomes, that is, the number of elements in the sample space.

P (A) = n (A) / n (Ω) = 3/8 = 0.375 = 37.5%

So there is a 37.5% probability of having two black marbles after randomly drawing three marbles from the bag. But note that we cannot in any way predict the exact outcome of the experiment.

Solution e

The probability that event B occurs, consisting of obtaining at least one black marble is:

P (B) = n (B) / n (Ω) = 4/8 = 0.5 = 50%

This means that the possibility that event B occurs is equal to the probability that it does not occur.

Solution f

The probability of obtaining at least one black marble, after drawing three of them, is equal to 1 minus the probability that the result will be "the three white marbles."

P (C) = 1 - P (bbb) = 1 - ⅛ = ⅞ = 0.875 = 87.5%

Now, we can check this result, noting that the number of possibilities that the event C occurs is equal to the number of elements of the possible results for the event C:

C = {(b, b, n), (b, n, b), (b, n, n), (n, b, b), (n, b, n), (n, n, b), (n, n, n)}

n (C) = 7

P (C) = n (C) / n (Ω) = ⅞ = 87.5%

References

1. CanalPhi. Random experiment. Recovered from: youtube.com.
2. MateMovil. Random experiment. Recovered from: youtube.com
3. Pishro Nick H. Introduction to probability. Recovered from: probabilitycourse.com
4. Ross. Probability and statistics for engineers. Mc-Graw Hill.
5. Wikipedia. Experiment (probability theory). Recovered from: en.wikipedia.com
6. Wikipedia. Deterministic event. Recovered from: es. wikipedia.com
7. Wikipedia. Random experiment. Recovered from: es.wikipedia.com