Conditional Probability and Dependent Events
You draw a card from a 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 they already know (the result) changes the probability of what they want to find out (whether there is a disease). In both cases, one event changes the probability of the other: they are dependent events, and you need conditional probability.
Conditional probability
When you already know that an event occurred, the set of possible cases shrinks. That new information updates the probability.
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?
We roll two standard six-sided dice. Given that the sum of the scores obtained is $8$, what is the probability that a $3$ was rolled on one of the dice?
We roll two standard six-sided dice. Given that the sum of the scores obtained is $8$, what is the probability that a $3$ was rolled on one of the dice?
In a town, $45\%$ of the inhabitants read newspaper X, $35\%$ read newspaper Y, and $15\%$ read both newspapers. If we choose an inhabitant at random and know that they…
In a town, $45\%$ of the inhabitants read newspaper X, $35\%$ read newspaper Y, and $15\%$ read both newspapers. If we choose an inhabitant at random and know that they read newspaper X, what is the probability that they also read newspaper Y? In light of the results, explain whether the events "reading newspaper X" and "reading newspaper Y" are independent.
Independence and union of events
If knowing that $A$ occurred does not change the probability of $B$, the events are independent: $P(B\mid A)=P(B)$, or equivalently $P(A\cap B)=P(A)\,P(B)$. If this does not hold, they are dependent.
Sometimes you are given $P(A\cup B)$ and need the intersection. When you add $P(A)+P(B)$, the intersection is counted twice, so you subtract it once:
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.
Let $A$ and $B$ be two events of the same random experiment such that $P(A) = 0.5$, $P(B) = 0.4$ and $P(A \cup B) = 0.7$. Using the properties of probability, find the…
Let $A$ and $B$ be two events of the same random experiment such that $P(A) = 0.5$, $P(B) = 0.4$ and $P(A \cup B) = 0.7$. Using the properties of probability, find the value of the intersection and calculate $P(A|B)$ and $P(B|A)$. Are $A$ and $B$ independent events? Explain your answer.
Product rule and draws without replacement
Try it with the urn: draw balls and watch how the makeup changes.
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 Spanish deck of $40$ cards, two cards are drawn at random one after the other and without replacement. What is the probability that both cards are kings?
From a Spanish deck of $40$ cards, two cards are drawn at random one after the other and without replacement. What is the probability that both cards are kings?
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?
Independent events: "at least one"
If the events are independent, the rule simplifies: $P(A\cap B)=P(A)\,P(B)$. And the probability that at least one occurs is best calculated with the complement: $1-P(\text{none})$.
A coin is biased such that the probability of getting Heads is $0.7$. The coin is tossed twice consecutively. Calculate the probability of getting exactly one Head and…
A coin is biased such that the probability of getting Heads is $0.7$. The coin is tossed twice consecutively. Calculate the probability of getting exactly one Head and one Tail (in any order).
Two archers shoot an arrow at the same target. Archer A hits the target with a probability of $0.7$ and Archer B hits the target with a probability of $0.8$. Their shots…
Two archers shoot an arrow at the same target. Archer A hits the target with a probability of $0.7$ and Archer B hits the target with a probability of $0.8$. Their shots are completely independent. What is the probability that the target is hit by at least one arrow?
A computer security system has two independent firewalls. The probability that firewall A fails during an attack is $0.15$ and the probability that firewall B fails is…
A computer security system has two independent firewalls. The probability that firewall A fails during an attack is $0.15$ and the probability that firewall B fails is $0.10$. The system is compromised only if both firewalls fail. What is the probability that the system withstands an attack? If a third firewall C, identical to B and independent of the previous ones, is added, by how much does the probability of the system withstanding the attack increase?
Trees and total probability
A tree diagram organizes the stages: each branch carries a conditional probability, and to reach the end of a path you multiply the branches. When several paths lead to the same outcome, you add.
Look at the following tree diagram representing a two-stage compound experiment. Calculate the probability of obtaining the final result $C$ going through branch $A$.
Look at the following tree diagram representing a two-stage compound experiment. Calculate the probability of obtaining the final result $C$ going through branch $A$.
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?
We have three chests that are externally identical. CHEST A contains $2$ gold coins and $1$ silver coins. CHEST B contains $1$ gold coin and $3$ silver coins. CHEST C…
We have three chests that are externally identical. CHEST A contains $2$ gold coins and $1$ silver coins. CHEST B contains $1$ gold coin and $3$ silver coins. CHEST C contains $4$ gold coins. We roll a fair die: if we get $1$ or $2$, we choose chest A; if we get $3$, we choose chest B; if we get $4$, $5$ or $6$, we choose chest C. Then, we draw a coin at random from the chosen chest. What is the probability that the drawn coin is gold?
Reversing the conditional: from an effect to its cause
A common question is the reverse one: if the test comes back positive, what is the real probability of being sick? This is the seed of Bayes' theorem.
On a shelf there are $5$ novels and $3$ poetry books. Two books are chosen at random one after the other and without replacement. If the second book chosen turned out to…
On a shelf there are $5$ novels and $3$ poetry books. Two books are chosen at random one after the other and without replacement. If the second book chosen turned out to be poetry, what is the probability that the first book chosen was also poetry?
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 beginning: if you draw two cards from a standard deck without putting the first one back, the probability that the second is a heart goes from $13/52=25\,\%$ to $12/51\approx23.5\,\%$ if the first one was a heart: the prior information changes the calculation.