There is talk of random experiment when the outcome of each particular trial is unpredictable, even though the probability of occurrence of a particular outcome 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..
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.
The toss of 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 100 ºC. But it never happens that, keeping the same conditions, the result is sometimes 90 ºC, other 12 0ºC and sometimes 100 ºC..
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The set of all possible outcomes of a random experiment is called 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, stamp.
In a random experiment, a event it is the occurrence or not of a certain result. For example, in the case of a coin toss, 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 the roll of 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.
Here are some examples that illustrate the above:
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 fact that the experiment results in 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.
a) It is a random experiment because there is no way to predict what will be the result of a toss of the two coins.
b) The sample space is the set of all possible outcomes:
S = (c, c), (c, s), (s, c), (s, s)
c) Event A, if it occurs, can have the following results:
A = (c, s), (s, c)
d) The probability that event A occurs is obtained by dividing the number of elements of set A by the number of elements of set S corresponding to the sample space:
P (A) = 2/4 = ½ = 0.5 = 50%
e) The set of possible outcomes corresponding to event B (not appearing heads in the result) is:
B = (s, s)
Therefore, the probability that event B occurs in a trial is the quotient between the number of possible outcomes of B and the number of total cases:
P (B) = ¼ = 0.25 = 25%.
A bag contains 10 white marbles and 10 black marbles. Three marbles consecutively are taken out of the bag at random and without looking inside.
a) Determine the sample space of 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.
To construct the sample space, it is useful to make a tree diagram, like the one shown in Figure 3:
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)
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)
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)
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% chance of having two black marbles after randomly drawing three marbles from the bag. But note that in no way can we predict the exact outcome of the experiment.
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.
The probability of obtaining at least one black marble, after extracting three of them, is equal to 1 minus the probability that the result is "the three white marbles".
P (C) = 1 - P (b b b) = 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%
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