Dependent and independent events
Whether it rains on Saturday and whether you pass Monday's exam have nothing to do with each other: knowing one does not change the probability of the other. But if the first card drawn from a pack is a king, that does change the probability that the second one is also a king. This section makes precise when two events are independent (they do not influence each other) and when they are dependent (they do), and how this simplifies — or not — the calculation of $P(A\cap B)$.
Independence of events
A 1st Baccalaureate student has two exams on the same day: Mathematics and English. The probability of passing Mathematics is $P(M) = 0,7$ and that of passing English is…
A 1st Baccalaureate student has two exams on the same day: Mathematics and English. The probability of passing Mathematics is $P(M) = 0,7$ and that of passing English is $P(I) = 0,8$. Assuming that passing one exam does not affect the result of the other:
- a) Calculate the probability of passing both exams.
- b) Calculate the probability of passing neither exam.
The product rule for independent events
When $A$ and $B$ are independent, the general product rule $P(A\cap B)=P(B)\cdot P(A|B)$ simplifies, because $P(A|B)=P(A)$.
Let $A$ and $B$ be two events associated with a random experiment. We know that the probability of event $A$ occurring is $P(A) = 0,6$ and the probability of event $B$…
Let $A$ and $B$ be two events associated with a random experiment. We know that the probability of event $A$ occurring is $P(A) = 0,6$ and the probability of event $B$ occurring is $P(B) = 0,3$. If the events are known to be independent, calculate the probability of both occurring at the same time, that is, $P(A \cap B)$.
An urn contains 5 yellow balls and 3 purple balls. We draw two balls from the urn consecutively and without replacement. a) Draw the tree diagram corresponding to this…
An urn contains 5 yellow balls and 3 purple balls. We draw two balls from the urn consecutively and without replacement.
- a) Draw the tree diagram corresponding to this experiment, indicating the probability of each branch.
- b) Calculate the probability that the two drawn balls are the same colour.
You can do it on paper and send a photo with your answer.
With replacement and independence; without replacement and dependence
Successive draws from an urn are the clearest example of when there is independence and when there is not: it depends on whether what was drawn is put back.
In a gumball machine there are balls of three colours: 10 red, 6 blue and 4 green. A child draws a gumball, notes its colour, puts it back into the machine and draws a…
In a gumball machine there are balls of three colours: 10 red, 6 blue and 4 green. A child draws a gumball, notes its colour, puts it back into the machine and draws a second gumball.
- a) Justify whether the events "drawing a red ball on the first" and "drawing a blue ball on the second" are dependent or independent.
- b) Calculate the probability that the first gumball is blue and the second is green.
We draw two cards from a Spanish 40-card deck, one after the other and without returning the first to the deck. a) What is the probability that the first card is a King?…
We draw two cards from a Spanish 40-card deck, one after the other and without returning the first to the deck.
- a) What is the probability that the first card is a King?
- b) Given that the first card was a King, what is the probability that the second is also a King?
- c) Are these events dependent or independent?
Independence of several events
With three or more events, it is not enough to check independence in pairs: the product rule must also work for all of them together.
Two electronic components of a drone, $A$ and $B$, function independently. The probability of component $A$ failing is $0,05$ and the probability of component $B$…
Two electronic components of a drone, $A$ and $B$, function independently. The probability of component $A$ failing is $0,05$ and the probability of component $B$ failing is $0,08$.
- a) Calculate the probability that the drone can keep flying if at least one of the two components must function correctly for this to happen.
- b) Calculate the probability that exactly one of the two components fails.
In a popular strategy video game, players open "legendary chests" to obtain rewards. The probability of getting an S-level hero from the first chest is only $0,05$.…
In a popular strategy video game, players open "legendary chests" to obtain rewards. The probability of getting an S-level hero from the first chest is only $0,05$. However, the game has an "accumulated luck" system: if you do not get the hero from a chest, the probability of getting it from the next one increases by $0,02$ compared with the probability for the previous chest. If you get the hero, the probability goes back to $0,05$ for the next attempt.
The figure shows the situation you must study: the first two chests do not give the hero and the hero appears in the third.
Calculate the probability that a player needs to open exactly three chests to get their first S-level hero.