Dependence
You draw a card from the deck and don't put it back: the second draw no longer has the same probabilities as the first. A doctor interprets the result of a test: what is already known (the test result) changes the probability of what we want to find out (whether the patient has the disease). In both cases, one event changes the probability of the other: they are dependent events, and you need conditional probability.
Conditional probability
Consider two events $A$ and $B$ of a random experiment. It is known that $P(A) = 0,4$, $P(B) = 0,5$ and $P(A \cap B) = 0,2$. Check mathematically whether the events $A$…
Consider two events $A$ and $B$ of a random experiment. It is known that $P(A) = 0,4$, $P(B) = 0,5$ and $P(A \cap B) = 0,2$. Check mathematically whether the events $A$ and $B$ are independent.
In a class of $30$ students, $18$ study French, $12$ study German and $5$ study both languages. If a student is chosen at random and turns out to study French, what is…
In a class of $30$ students, $18$ study French, $12$ study German and $5$ study both languages. If a student is chosen at random and turns out to study French, what is the probability that they also study German?
The product rule and draws without replacement
An urn contains $5$ red balls and $3$ blue balls. Two balls are drawn consecutively and without replacement. Calculate the probability that the first ball is red and the…
An urn contains $5$ red balls and $3$ blue balls. Two balls are drawn consecutively and without replacement. Calculate the probability that the first ball is red and the second is blue.
From a box containing $10$ light bulbs, it is known that $3$ of them are blown. $3$ light bulbs are drawn successively without replacement. What is the probability that…
From a box containing $10$ light bulbs, it is known that $3$ of them are blown. $3$ light bulbs are drawn successively without replacement. What is the probability that exactly one of the drawn bulbs is blown?
Total probability: adding up several paths
In a factory, Machine A produces $60\%$ of the parts and Machine B produces the remaining $40\%$. It is known that $2\%$ of the parts produced by A are defective, while…
In a factory, Machine A produces $60\%$ of the parts and Machine B produces the remaining $40\%$. It is known that $2\%$ of the parts produced by A are defective, while for Machine B the percentage of defective parts is $5\%$. If a part is chosen at random from the warehouse, what is the probability that it was manufactured by Machine B and is defective?
A rapid allergy test is carried out in a clinic. It is known that $8\%$ of the patients are allergic to a certain substance. The test gives a positive result in $90\%$…
A rapid allergy test is carried out in a clinic. It is known that $8\%$ of the patients are allergic to a certain substance. The test gives a positive result in $90\%$ of the patients who are actually allergic. However, it also gives a false positive in $5\%$ of the patients who are not allergic. A patient is chosen at random and tested. What is the probability that the result is positive?
Inverting conditional probability: from an effect back to its cause
A common question is the reverse of conditional probability: if the test comes back positive, what is the real probability of having the disease? This is the seed of Bayes' theorem, which you will see in full detail, with tree diagrams, in the next section.
In a school, $55\%$ of the students are girls. It is known that $20\%$ of the girls and $25\%$ of the boys actively participate in the drama club. If a student who…
In a school, $55\%$ of the students are girls. It is known that $20\%$ of the girls and $25\%$ of the boys actively participate in the drama club. If a student who participates in the drama club is chosen at random, what is the probability that they are a girl?
We have two cubic dice in a shaker. The first is a standard die with faces numbered from $1$ to $6$. The second is a biased die that has three faces with a $2$, two…
We have two cubic dice in a shaker. The first is a standard die with faces numbered from $1$ to $6$. The second is a biased die that has three faces with a $2$, two faces with a $3$ and one face with a $5$. A die is chosen at random (with a probability of $\frac{1}{2}$ for each) and rolled. If the score obtained is an even number, what is the probability that the biased die was rolled?
Back to the start: If you draw two cards from a deck of $40$ without putting the first one back, the probability of drawing a cup (copas) on the second draw goes from $10/40$ ($25\,\%$) to just $9/39$ (approximately $23{,}1\,\%$) if the first one was a cup.