Basic concepts
In 4th ESO you already worked with events and probabilities. In Bachillerato tecnológico you will need the same idea but with more precision: a control system, a cryptography algorithm or a reliability analysis does not allow "more or less". Before defining probability rigorously (you will do that in the next section), we need to fix the language: what exactly an event is and how events are combined with each other, with the same precision as an operation with sets.
Random experiment and sample space
An $8$-sided die in the shape of a regular octahedron, numbered from $1$ to $8$, is rolled. Write down the sample space $E$ associated with this random experiment and…
An $8$-sided die in the shape of a regular octahedron, numbered from $1$ to $8$, is rolled. Write down the sample space $E$ associated with this random experiment and determine the outcomes that make up the following events:
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$A =$ "Obtaining a prime number".
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$B =$ "Obtaining a number greater than $5$".
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$C = A \cap B$.
$3$ fair coins are tossed simultaneously. Write down the complete sample space $E$ using the notation $C$ for heads and $X$ for tails. Next, calculate the probability of…
$3$ fair coins are tossed simultaneously. Write down the complete sample space $E$ using the notation $C$ for heads and $X$ for tails. Next, calculate the probability of getting exactly two heads.
Operations with events: union, intersection, complement
Since events are sets, they are combined with the same operations as sets: probability theory does not invent anything new here, it just reuses the algebra of sets that you already know.
| Operation | Notation | Meaning |
|---|---|---|
| Union | $A \cup B$ | $A$ occurs, or $B$, or both |
| Intersection | $A \cap B$ | both $A$ and $B$ occur at the same time |
| Complement | $\overline{A}$ | $A$ does not occur (it is $E - A$) |
| Difference | $A - B$ | $A$ occurs but not $B$ (it is $A \cap \overline{B}$) |
In a technology college with $150$ students, a survey was carried out on the use of programming languages. The results show that $85$ students program in Python ($P$),…
In a technology college with $150$ students, a survey was carried out on the use of programming languages. The results show that $85$ students program in Python ($P$), $60$ program in Java ($J$) and $25$ program in both languages.
If a student is selected at random from this college, find the following probabilities:
- a) That they program in at least one of the two languages.
- b) That they do not program in either of these two languages.
- c) That they program in Python, but not in Java.
De Morgan's laws
Negating a union or an intersection follows a fixed rule: it is the same idea you will use in logic circuits and in programming with AND/OR/NOT.
In a meteorological study, the probability that it rains on Saturday is $P(S)$ and that it rains on Sunday is $P(D)$. Knowing that the probability that it rains on at…
In a meteorological study, the probability that it rains on Saturday is $P(S)$ and that it rains on Sunday is $P(D)$. Knowing that the probability that it rains on at least one of the two days of the weekend is $P(S \cup D) = 0,85$, use De Morgan's laws or the properties of the complementary event to calculate the probability that it does not rain on either day of the weekend, that is, $P(\overline{S} \cap \overline{D})$.
Compatible and incompatible events
The random experiment of rolling a fair cubic die numbered from $1$ to $6$ is performed. Consider the events $A =$ "Obtaining an even number" and $B =$ "Obtaining a…
The random experiment of rolling a fair cubic die numbered from $1$ to $6$ is performed. Consider the events $A =$ "Obtaining an even number" and $B =$ "Obtaining a prime number greater than $2$".
Write down the outcomes of each event and determine, justifying your answer, whether events $A$ and $B$ are compatible or incompatible.
In an environmental toxicology study in a river, three pollutants are monitored. Let $A$, $B$ and $C$ be the events of finding dangerous levels of the respective…
In an environmental toxicology study in a river, three pollutants are monitored. Let $A$, $B$ and $C$ be the events of finding dangerous levels of the respective pollutant. It is known that $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{3}$ and $P(C) = \frac{1}{4}$.
Furthermore, it is known that pollutants $A$ and $B$ never appear together in dangerous levels (they are incompatible events), and that whenever pollutant $C$ is detected, pollutant $B$ is also detected (that is, $C$ is a subset of $B$, $C \subset B$).
Calculate the probability of finding neither pollutant $A$ nor pollutant $B$, that is, $P(\overline{A} \cap \overline{B})$, and the probability of finding at least one of the three pollutants, $P(A \cup B \cup C)$.
The union of three events
With three events at once, adding the three probabilities counts the regions where two or all three overlap too many times: we have to correct this by subtracting the pairwise intersections and adding back the triple intersection.
On a digital entertainment platform with $500$ subscribers, preferences for three genres are analysed: Action ($A$), Comedy ($C$) and Drama ($D$). The data reveal that:…
On a digital entertainment platform with $500$ subscribers, preferences for three genres are analysed: Action ($A$), Comedy ($C$) and Drama ($D$). The data reveal that:
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$220$ watch Action, $180$ watch Comedy, $150$ watch Drama.
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$70$ watch Action and Comedy, $50$ watch Action and Drama, $40$ watch Comedy and Drama.
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$15$ watch all three genres.
If a subscriber is chosen at random, find the probability that they:
- a) Watch Action movies exclusively.
- b) Watch exactly one of the three genres.
- c) Watch none of these three genres.