Definitions
Until now, probability has been used intuitively: "favourable outcomes over possible outcomes". But that rule only works if all the outcomes are equally likely, and it does not explain why $P(\overline{A}) = 1-P(A)$ or why $P(A \cup B) = P(A)+P(B)-P(A\cap B)$ always work. The mathematician Andrey Kolmogorov formalised probability in 1933 with just three starting rules (the axioms); everything else, including Laplace's rule, is deduced from them.
Probability as a function: Kolmogorov's axioms
Let $A$ and $B$ be two events in the same random experiment such that $P(A) = 0,4$, $P(B) = 0,3$ and $P(A \cup B) = 0,7$. Justify mathematically whether events $A$ and…
Let $A$ and $B$ be two events in the same random experiment such that $P(A) = 0,4$, $P(B) = 0,3$ and $P(A \cup B) = 0,7$. Justify mathematically whether events $A$ and $B$ are compatible or incompatible based on the axioms of probability.
Laplace's rule: a particular case that satisfies the axioms
Laplace's rule is not a separate law: it is the form that $P$ takes when the $n$ outcomes of $E$ are equally likely. Checking that it satisfies the three axioms is easy, and it gives confidence in why it works.
An opaque urn contains 5 red balls, 3 blue balls and 2 green balls, all identical to the touch. If a ball is drawn at random, calculate the probability of: Drawing a…
An opaque urn contains 5 red balls, 3 blue balls and 2 green balls, all identical to the touch. If a ball is drawn at random, calculate the probability of:
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Drawing a blue ball.
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Not drawing a red ball.
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Drawing a ball that is red or green.
Properties deduced from the axioms
None of these rules is a new axiom: they can all be proved from Kolmogorov's three axioms, and that is why they are always valid, whether the outcomes are equally likely or not.
In an electronic components factory, it is known that the probability of a chip having a soldering defect is $0,05$, the probability of having a packaging defect is…
In an electronic components factory, it is known that the probability of a chip having a soldering defect is $0,05$, the probability of having a packaging defect is $0,03$, and the probability of having both defects simultaneously is $0,01$. Calculate the probability that a randomly chosen chip has at least one of the two defects.
In a software development company, the probability that a program has a user interface error (event $U$) is $0,45$. The probability that it has a database error (event…
In a software development company, the probability that a program has a user interface error (event $U$) is $0,45$. The probability that it has a database error (event $D$) is $0,30$. If the probability that it has a user interface error but not a database error is $0,25$, calculate the probability that the program has no errors.
Venn diagrams with areas proportional to probability
A Venn diagram can be drawn so that the area of each region is proportional to its probability: it is a visual way of checking that the calculations are consistent with the axioms (the whole of $E$ has area $1$).
Look at the following Venn diagram representing a sample space $E$ and two events $A$ and $B$. Knowing that the area of each region is proportional to its probability,…
Look at the following Venn diagram representing a sample space $E$ and two events $A$ and $B$. Knowing that the area of each region is proportional to its probability, express the region shaded in green in set notation and calculate its probability if $P(A) = 0,55$, $P(B) = 0,40$ and $P(A \cap B) = 0,15$.
In a school, $60\%$ of 1st Baccalaureate students take Physics, $50\%$ take Technical Drawing and $20\%$ take neither of the two. If a student is chosen at random,…
In a school, $60\%$ of 1st Baccalaureate students take Physics, $50\%$ take Technical Drawing and $20\%$ take neither of the two. If a student is chosen at random, determine the following probabilities using axiomatic properties and De Morgan's Laws:
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That they take both subjects.
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That they take Physics but not Technical Drawing.
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That they take exactly one of the two subjects.
Bounds on the probability when the intersection is unknown
The axioms do not only give equalities: they also let us bound an unknown probability when information is missing, using the fact that no probability can exceed $1$ or be negative.
In a quality control check, two tests are applied to a batch of products. It is known that the probability of passing test $A$ is $P(A) = 0,85$ and the probability of…
In a quality control check, two tests are applied to a batch of products. It is known that the probability of passing test $A$ is $P(A) = 0,85$ and the probability of passing test $B$ is $P(B) = 0,75$. Without knowing the probability of the intersection, determine algebraically the maximum value and the minimum value that the probability of a product passing both tests, $P(A \cap B)$, can take.
Estimating probabilities using frequencies
When we do not know whether a mechanism is fair, it is not enough to count outcomes and apply Laplace's rule. We repeat the experiment under comparable conditions and estimate $P(A)$ with the relative frequency $f_n(A)=n_A/n$.
With the urn of 5 red balls and 5 blue balls in the simulator, each draw is made with replacement. Reset it and record the number of reds and their frequency on reaching 20, 200 and 2000 trials. Use the batches of 1, 10, 100 and 1000 to reach those totals. Compare with $P(\text{red})=0{,}5$ and repeat the whole experiment.