Independence
Does getting heads on a coin change the probability of rolling a $5$ on a die? No. That is why you can simply multiply their probabilities. But in a problem with data and tables it is not always obvious whether two events are independent: you have to check it with a precise mathematical condition, not "by eye".
Formal definition of independence
Let $A$ and $B$ be two events in a random experiment. It is known that $P(A) = 0,4$, $P(B) = 0,5$ and $P(A \cap B) = 0,2$. Determine algebraically whether the events $A$…
Let $A$ and $B$ be two events in a random experiment. It is known that $P(A) = 0,4$, $P(B) = 0,5$ and $P(A \cap B) = 0,2$. Determine algebraically whether the events $A$ and $B$ are independent.
In a statistical study on reading habits, $200$ people were surveyed. The results show that $120$ read regularly and $80$ do not. Furthermore, $90$ of the respondents…
In a statistical study on reading habits, $200$ people were surveyed. The results show that $120$ read regularly and $80$ do not. Furthermore, $90$ of the respondents have a university degree, of whom $54$ read regularly. Test mathematically whether the events "reading regularly" and "having a university degree" are independent in this sample.
Checking independence from data
When the data come from a table, a survey or a statement with percentages, you first have to calculate the three probabilities ($P(A)$, $P(B)$, $P(A\cap B)$) from the counts, and then compare them.
Two events $M$ and $N$ are incompatible (mutually exclusive) and have probabilities strictly greater than zero. Can they be independent? Justify your answer using the…
Two events $M$ and $N$ are incompatible (mutually exclusive) and have probabilities strictly greater than zero. Can they be independent? Justify your answer using the mathematical definition of independence.
Consider two events $A$ and $B$ from the same sample space such that $P(A) = 0,5$, $P(B) = 0,4$ and $P(A \cup B) = 0,8$. Show algebraically that events $A$ and $B$ are…
Consider two events $A$ and $B$ from the same sample space such that $P(A) = 0,5$, $P(B) = 0,4$ and $P(A \cup B) = 0,8$. Show algebraically that events $A$ and $B$ are not independent, and then calculate the conditional probability $P(A|B)$.
Independence and the probability of the union
Let $A$ and $B$ be two independent events associated with a random experiment. It is known that the probability of the union is $P(A \cup B) = 0,88$ and the probability…
Let $A$ and $B$ be two independent events associated with a random experiment. It is known that the probability of the union is $P(A \cup B) = 0,88$ and the probability of $A$ is $P(A) = 0,6$. Calculate the probability of event $B$.
Independence in systems with several components
When several components (sensors, firewalls, players) act separately, without influencing one another, the product extends to as many factors as there are components.
A high-tech bank security system consists of three motion sensors operating completely independently. The probabilities of each sensor detecting an intruder entering the…
A high-tech bank security system consists of three motion sensors operating completely independently. The probabilities of each sensor detecting an intruder entering the vault are $0,90$, $0,85$ and $0,95$, respectively. Calculate the probability that the system detects an intruder (that is, that at least one sensor is activated).
Prove analytically that if two events $A$ and $B$ are independent, then their complementary events $\overline{A}$ and $\overline{B}$ are also independent. To do so, use…
Prove analytically that if two events $A$ and $B$ are independent, then their complementary events $\overline{A}$ and $\overline{B}$ are also independent. To do so, use the algebraic properties of probability and the fundamental definition of independence.