INDEPENDENT EVENTS: DEMONSTRATION, EXAMPLES, EXERCISES - DUDAS - 2025

Two events are independent, when the probability that one of them occurs is not influenced by the fact that the other occurs -or does not occur-, considering that these events occur randomly.

This circumstance occurs whenever the process that generates the result of event 1, does not alter in any way the probability of the possible results of event 2. But if this does not happen, the events are said to be dependent.

Figure 1. Colored marbles are frequently used to explain the probability of independent events. Source: Pixabay.

An independent event situation is as follows: Suppose two six-sided dice are rolled, one blue and the other pink. The probability that a 1 will roll on the blue die is independent of the probability that a 1 will roll -or not roll- on the pink die.

Another case of two independent events is that of tossing a coin twice in a row. The result of the first throw will not depend on the result of the second and vice versa.

Proof of two independent events

To verify that two events are independent, we will define the concept of conditional probability of one event with respect to another. For this, it is necessary to differentiate between exclusive events and inclusive events:

Two events are exclusive if the possible values ​​or elements of event A have nothing in common with the values ​​or elements of event B.

Therefore in two exclusive events, the set of the intersection of A with B is the vacuum:

Excluding events: A∩B = Ø

On the contrary, if the events are inclusive, it may happen that a result of event A also coincides with that of another B, with A and B being different events. In this case:

Inclusive events: A∩B ≠ Ø

This leads us to define the conditional probability of two inclusive events, in other words, the probability of occurrence of event A, whenever event B occurs:

P (A¦B) = P (A∩B) / P (B)

Therefore, the conditional probability is the probability that A and B will occur divided by the probability that B will occur. The probability that B will occur conditional on A can also be defined:

P (B¦A) = P (A∩B) / P (A)

Criteria to know if two events are independent

Next we will give three criteria to know if two events are independent. It is enough that one of the three is fulfilled, so that the independence of events is demonstrated.

1.- If the probability that A occurs whenever B occurs is equal to the probability of A, then they are independent events:

P (A¦B) = P (A) => A is independent of B

2.- If the probability that B occurs given A, is equal to the probability of B, then there are independent events:

P (B¦A) = P (B) => B is independent of A

3.- If the probability that A and B occur is equal to the product of the probability that A occurs and the probability that B occurs, then they are independent events. The converse is also true.

P (A∩B) = P (A) P (B) <=> A and B are independent events.

Examples of independent events

Rubber soles produced by two different suppliers are compared. The samples from each manufacturer are subjected to several tests from which it is concluded whether or not they are within the specifications.

Figure 2. Variety of rubber soles. Source: Pixabay.

The resulting summary of the 252 samples is as follows:

Manufacturer 1; 160 do meet specifications; 8 do not meet specifications.

Manufacturer 2; 80 do meet specifications; 4 do not meet specifications.

Event A: "that the sample is from manufacturer 1".

Event B: "that the sample meets the specifications."

We want to know if these events A and B are independent or not, for which we apply one of the three criteria mentioned in the previous section.

Criterion: P (B¦A) = P (B) => B is independent of A

P (B) = 240/252 = 0.9523

P (B¦A) = P (A ⋂ B) / P (A) = (160/252) / (168/252) = 0.9523

Conclusion: Events A and B are independent.

Suppose event C: "that the sample comes from manufacturer 2"

Will event B be independent of event C?

We apply one of the criteria.

Criterion: P (B¦C) = P (B) => B is independent of C

P (B¦C) = (80/252) / (84/252) = 0.9523 = P (B)

Therefore, based on available data, the probability that a randomly chosen rubber sole meets specifications is independent of the manufacturer.

Convert an Independent Event to a Dependent Event

Let's look at the following example to distinguish between dependent and independent events.

We have a bag with two white chocolate balls and two black balls. The probability of getting a white ball or a black ball is equal on the first try.

Suppose the result was a cue ball. If the drawn ball is replaced in the bag, the original situation is repeated: two white balls and two black balls.

So in a second event or draw, the chances of drawing a cue ball or a black ball are identical to the first time. They are therefore independent events.

But if the cue ball drawn in the first event is not replaced because we have eaten it, in the second draw there are greater chances of drawing a black ball. The probability that a second extraction will obtain white again is different from that of the first event and is conditioned by the previous result.

Exercises

- Exercise 1

In a box we put the 10 marbles of figure 1, of which 2 are green, 4 are blue and 4 are white. Two marbles will be chosen at random, one first and one later. It is asked to find the

probability that none of them is blue, under the following conditions:

a) With replacement, that is, returning the first marble before the second selection to the box. Indicate whether they are independent or dependent events.

b) Without replacement, in such a way that the first marble extracted is left out of the box at the time of making the second selection. Similarly, indicate whether they are dependent or independent events.

Solution to

We calculate the probability that the first marble extracted is not blue, which is 1 minus the probability that it is blue P (A), or directly that it is not blue, because it came out green or white:

P (A) = 4/10 = 2/5

P (don't be blue) = 1 - (2/5) = 3/5

O well:

P (green or white) = 6/10 = 3/5.

If the extracted marble is returned, everything is as before. In this second draw there is also a 3/5 probability that the marble drawn is not blue.

P (not blue, not blue) = (3/5). (3/5) = 9/25.

The events are independent, since the extracted marble was returned to the box and the first event does not influence the probability of occurrence of the second.

Solution b

For the first extraction, proceed as in the previous section. The probability that it is not blue is 3/5.

For the second extraction we have 9 marbles in the bag, since the first did not return, but it was not blue, therefore in the bag there are 9 marbles and 5 not blue:

P (green or white) = 5/9.

P (none is blue) = P (first not blue). P (second not blue / first not blue) = (3/5). (5/9) = 1/3

In this case they are not independent events, since the first event conditions the second.

- Exercise 2

A store has 15 shirts in three sizes: 3 small, 6 medium and 6 large. 2 shirts are randomly selected.

a) What is the probability that both shirts selected are small, if one is taken first and without replacing another in the lot?

b) What is the probability that both selected shirts are small, if one is drawn first, is replaced in the batch, and the second is removed?

Solution to

Here are two events:

Event A: the first shirt selected is small

Event B: the second selected shirt is small

The probability that event A occurs is: P (A) = 3/15

The probability that event B occurs is: P (B) = 2/14, because a shirt had already been removed (there are 14 left), but also the event A wants to be fulfilled, the first shirt removed must be small and therefore both are 2 small.

That is, the probability that A and B will be the product of the probabilities is:

P (A and B) = P (B¦A) P (A) = (2/14) (3/15) = 0.029

Therefore, the probability that event A and B occurs is equal to the product that event A occurs, times the probability that event B occurs if event A.

It should be noted that:

P (B¦A) = 2/14

The probability that event B occurs regardless of whether event A occurs or not will be:

P (B) = (2/14) if the first was small, or P (B) = 3/14 if the first was not small.

In general, the following can be concluded:

P (B¦A) is not equal to P (B) => B is not independent of A

Solution b

Again there are two events:

Event A: the first shirt selected is small

Event B: the second selected shirt is small

P (A) = 3/15

Remember that whatever the result, the shirt drawn from the batch is replaced and again a shirt is drawn at random. The probability that event B occurs, if event A occurred is:

P (B¦A) = 3/15

The probability that events A and B occur will be:

P (A and B) = P (B¦A) P (A) = (3/15) (3/15) = 0.04

Note that:

P (B¦A) is equal to P (B) => B is independent of A.

- Exercise 3

Consider two independent events A and B. It is known that the probability that event A occurs is 0.2 and the probability that event B occurs is 0.3. What is the probability that both events occur?

Solution 2

Knowing that the events are independent, it is known that the probability that both events occur is the product of the individual probabilities. That is to say, P (A∩B) = P (A) P (B) = 0.2 * 0.3 = 0.06

Note that it is a probability far less than the probability that each event will occur regardless of the outcome of the other. Or put another way, much lower than the individual odds.

References

1. Berenson, M. 1985. Statistics for management and economics. Interamericana SA 126-127.
2. Monterrey Institute. Probability of independent events. Recovered from: monterreyinstitute.org
3. Math teacher. Independent events. Recovered from: youtube.com
4. Superprof. Types of events, dependent events. Recovered from: superprof.es
5. Virtual tutor. Probability. Recovered from: vitutor.net
6. Wikipedia. Independence (probability). Recovered from: wikipedia.com