Dependence
Seán picks a card from a standard deck of 52 playing cards at the school Christmas party and doesn'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
Independent events
If $B$ is not affected by $A$, then $P(B \mid A) = P(B)$ and the product rule becomes $P(A \cap B) = P(A) \cdot P(B)$. The events are independent. This is also the test: $A$ and $B$ are independent exactly when $P(A \cap B) = P(A)\cdot P(B)$. To check, work out $P(A)\cdot P(B)$ and compare it with $P(A \cap B)$: if they are equal, the events are independent; if not, they are dependent.
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.
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 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?
A card is drawn from a pack of cards, and then a coin is tossed. Are these two events dependent or independent? Calculate the probability of getting an "Ace" from the…
A card is drawn from a pack of cards, and then a coin is tossed. Are these two events dependent or independent? Calculate the probability of getting an "Ace" from the deck and "Heads" on the coin.
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?
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.
The product rule and draws without replacement
A standard deck has $52$ cards, with $4$ kings. The same method works for any deck: use the total number of cards, and then one fewer, in the denominators.
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 pack of $40$ cards (four suits of ten cards each), two cards are drawn at random one after the other and without replacement. What is the probability that both…
From a pack of $40$ cards (four suits of ten cards each), two cards are drawn at random one after the other and without replacement. What is the probability that both cards are kings?
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 $A$.
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 $52$ without putting the first one back, the probability of drawing a heart on the second draw goes from $13/52$ ($25\,\%$) to just $12/51$ (approximately $23.5\,\%$) if the first one was a heart.