Independent Events
Does flipping heads on a coin change your chance of rolling a $5$ on a die? No, so you can just multiply the two probabilities. But in a problem with a survey or a two-way table, it is not always obvious whether two events are independent. You need a precise math test, not a gut feeling. By the end of this page you will know that test and how to use it.
The definition of independence
Why does it work? If $A$ happening doesn't change the chance of $B$, then $B$ happens in the same fraction of cases whether or not $A$ occurred. So the fraction of the time that both happen is the fraction for $A$ times the fraction for $B$.
Independence and conditional probability
The notation $P(B \mid A)$ means "the probability of $B$ given that $A$ has occurred." It is defined by
$$P(B \mid A) = \frac{P(A \cap B)}{P(A)}$$Now divide the independence equation by $P(A)$. You get an equivalent way to say the same thing:
So if a problem says $G$ and $H$ are independent and gives $P(H \mid G) = 0.4$, then $P(H) = 0.4$ too. The condition changes nothing.
Let $A$ and $B$ be two events of a random experiment. It is known that $P(A) = 0,4$, $P(B) = 0,5$ and $P(A \cap B) = 0,2$. Determine analytically if events $A$ and $B$…
Let $A$ and $B$ be two events of a random experiment. It is known that $P(A) = 0,4$, $P(B) = 0,5$ and $P(A \cap B) = 0,2$. Determine analytically if events $A$ and $B$ are independent.
It is known that the probability of event $G$ occurring is $P(G) = 0,7$ and the probability of event $H$ occurring given that $G$ has occurred is $P(H|G) = 0,4$. If $G$…
It is known that the probability of event $G$ occurring is $P(G) = 0,7$ and the probability of event $H$ occurring given that $G$ has occurred is $P(H|G) = 0,4$. If $G$ and $H$ are independent, what is the value of $P(H)$?
Checking independence from data
When the information comes from a table, a survey, or percentages, first compute the three probabilities $P(A)$, $P(B)$ and $P(A \cap B)$ from the counts. Then compare $P(A)\cdot P(B)$ with $P(A \cap B)$.
In a statistical study on reading habits, $200$ people have been surveyed. The results show that $120$ read regularly and $80$ do not. Furthermore, $90$ of the…
In a statistical study on reading habits, $200$ people have been surveyed. The results show that $120$ read regularly and $80$ do not. Furthermore, $90$ of the respondents have a college degree, of which $54$ read regularly. Check mathematically if the events "reading regularly" and "having a college degree" are independent in this sample.
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 $B$ of the same sample space such that $P(A) = 0,5$, $P(B) = 0,4$ and $P(A \cup B) = 0,8$. Prove analytically that events $A$ and $B$ are not…
Consider two events $A$ and $B$ of the same sample space such that $P(A) = 0,5$, $P(B) = 0,4$ and $P(A \cup B) = 0,8$. Prove analytically that events $A$ and $B$ are not independent and, then, calculate the conditional probability $P(A|B)$.
The multiplication rule
Once you know two events are independent, the definition works as a tool: it gives you the probability that both happen.
The same equation also lets you work backward. If you know the intersection and one probability, divide to find the other.
Multiplying along a tree diagram
In a tree diagram for two independent stages, you multiply the probabilities along a path. The second stage has the same branch probabilities no matter what happened in the first.
In a random experiment, events $C$ and $D$ are independent. If $P(C) = 0,6$ and $P(D) = 0,3$, calculate the probability that both events occur at the same time, that is,…
In a random experiment, events $C$ and $D$ are independent. If $P(C) = 0,6$ and $P(D) = 0,3$, calculate the probability that both events occur at the same time, that is, find $P(C \cap D)$.
Events $E$ and $F$ are independent. Knowing that the probability of their intersection is $P(E \cap F) = 0,15$ and that $P(E) = 0,3$, find the probability of event $F$.
Events $E$ and $F$ are independent. Knowing that the probability of their intersection is $P(E \cap F) = 0,15$ and that $P(E) = 0,3$, find the probability of event $F$.
Observe the following tree diagram that models two independent experiments. Calculate the probability that event $\overline{A}$ occurs in the first stage and event $B$…
Observe the following tree diagram that models two independent experiments. Calculate the probability that event $\overline{A}$ occurs in the first stage and event $B$ in the second stage.
Independence in real situations
The multiplication rule works for anything independent: two coins, a coin and a die, two shooters, two separate genes in a plant cross. If the probability of each part is known, just multiply.
A card is drawn from a Spanish deck of $40$ cards, its suit is noted, it is returned to the deck, shuffled, and a second card is drawn. Calculate the probability that…
A card is drawn from a Spanish deck of $40$ cards, its suit is noted, it is returned to the deck, shuffled, and a second card is drawn. Calculate the probability that both drawn cards are of the "coins" suit.
In an archery competition, archer Alpha has a probability of $0,8$ of hitting the center of the target, while archer Beta has a probability of $0,7$. If both shoot an…
In an archery competition, archer Alpha has a probability of $0,8$ of hitting the center of the target, while archer Beta has a probability of $0,7$. If both shoot an arrow completely independently, calculate the probability that both hit the center of the target.
In a Mendelian genetic cross, seed color (Yellow dominant, green recessive) and seed texture (Smooth dominant, wrinkled recessive) are inherited completely…
In a Mendelian genetic cross, seed color (Yellow dominant, green recessive) and seed texture (Smooth dominant, wrinkled recessive) are inherited completely independently. If two heterozygous plants for both traits are crossed, the probability of inheriting the recessive allele from each parent for a single gene is $0,5$. Calculate the probability that a randomly chosen offspring plant has green and wrinkled seeds simultaneously.
"At least one" and the union
What if you want the probability that $A$ or $B$ happens (or both)? The general formula is $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. We subtract the overlap so it isn't counted twice. For independent events, the overlap is $P(A)\cdot P(B)$.
A second approach is often faster. "At least one" is the opposite of "none," so
$$P(\text{at least one}) = 1 - P(\text{none})$$and for independent events $P(\text{none})$ is a product of the complements.
In a coastal city, the probability that it rains on a Saturday is $40\%$, and the probability that it rains on a Sunday is $30\%$. Assuming that Sunday's weather is…
In a coastal city, the probability that it rains on a Saturday is $40\%$, and the probability that it rains on a Sunday is $30\%$. Assuming that Sunday's weather is independent of Saturday's weather, find the probability that it rains during the weekend (that is, it rains on Saturday, Sunday or both days).
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 $B$.
A high-tech bank security system consists of three motion sensors that operate completely independently. The probabilities that each sensor detects an intruder entering…
A high-tech bank security system consists of three motion sensors that operate completely independently. The probabilities that each sensor detects 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).
Combining cases: "exactly one" and "same outcome"
Many problems need more than one product. The plan is the same every time: list the different ways the event can happen, multiply within each way (independence), then add the ways (they can't happen at the same time).
A student has to take a multiple-choice exam. The probability that they know the correct answer to a history question is $0,7$, and the probability that they know the…
A student has to take a multiple-choice exam. The probability that they know the correct answer to a history question is $0,7$, and the probability that they know the answer to a geography question is $0,8$. If they are asked one question from each subject and their knowledge of them is independent, what is the probability that they know how to answer exactly one of the two questions?
There are two urns with the composition shown in the figure. A ball is drawn at random from urn $A$ and, independently, another ball from urn $B$. Calculate the…
There are two urns with the composition shown in the figure. A ball is drawn at random from urn $A$ and, independently, another ball from urn $B$. Calculate the probability that both drawn balls are the same color.
Systems and taking turns
A system often works only if certain parts work. Break it into pieces. "A and (B or C)" means: multiply $P(A \text{ works})$ by $P(\text{at least one of B, C works})$, and use the complement for that second part.
Taking turns
Games where players alternate turns can be handled the same way. Each turn is independent, and a win on a later turn means everyone before failed.
In a hospital, a patient undergoes two independent diagnostic tests to detect a respiratory allergy. Test $X$ gives a false negative (it does not detect the allergy when…
In a hospital, a patient undergoes two independent diagnostic tests to detect a respiratory allergy. Test $X$ gives a false negative (it does not detect the allergy when the patient does have it) with a probability of $0,15$. Test $Y$ gives a false negative with a probability of $0,10$. If the tested patient is truly allergic, what is the probability that neither test detects their allergy?
An electronic circuit for a communications satellite is formed by three components $A$, $B$ and $C$ arranged as shown in the diagram. The circuit works correctly if…
An electronic circuit for a communications satellite is formed by three components $A$, $B$ and $C$ arranged as shown in the diagram. The circuit works correctly if component $A$ works AND, in addition, at least one of the components $B$ or $C$ works. The three components fail independently with probabilities $P(F_A) = 0,02$, $P(F_B) = 0,1$ and $P(F_C) = 0,1$. Calculate the probability that the circuit as a whole works correctly.
Two friends, Ashley and Chloe, alternately roll a fair six-sided die in a game. The first player to get a $6$ wins the game. If Ashley has the first turn and they roll…
Two friends, Ashley and Chloe, alternately roll a fair six-sided die in a game. The first player to get a $6$ wins the game. If Ashley has the first turn and they roll independently, calculate the exact probability that Ashley wins the game on her first or second roll.
Complements and replacement
Complements of independent events
If $A$ and $B$ are independent, so are $\bar A$ and $\bar B$ (and $A$ with $\bar B$, and $\bar A$ with $B$). This means "neither" is a simple product:
$$P(\text{neither } A \text{ nor } B) = P(\bar A)\cdot P(\bar B) = (1 - P(A))(1 - P(B))$$Proving things with the definition
To prove two events are independent, show $P(\text{both}) = P(\text{first}) \cdot P(\text{second})$, using only algebra and the probability rules you know.
The same kind of argument works for $A$ and $\bar B$: start from $P(A \cap \bar B) = P(A) - P(A \cap B) = P(A) - P(A)P(B) = P(A)\big(1 - P(B)\big)$.
With and without replacement
Drawing with replacement (put the item back) makes the draws independent. Drawing without replacement changes what is left, so the draws are generally dependent.
A language academy offers French and German classes. $60\%$ of the students study French. It is known that the events "studying French" and "studying German" are…
A language academy offers French and German classes. $60\%$ of the students study French. It is known that the events "studying French" and "studying German" are completely independent. If the probability that a randomly chosen student studies neither of the two languages is $12\%$, determine what percentage of the academy's students study German.
A box contains $3$ green tokens (V) and $2$ orange tokens (N). Two consecutive draws are going to be made. Analyze the tree diagrams corresponding to performing the…
A box contains $3$ green tokens (V) and $2$ orange tokens (N). Two consecutive draws are going to be made. Analyze the tree diagrams corresponding to performing the experiment with replacement (Panel A) and without replacement (Panel B). Calculate the probability of drawing a green token on the second draw in both cases and explain mathematically in which of the two scenarios the result of the second draw is independent of the result of the first.
Prove analytically that if two events $A$ and $B$ are independent, then their complementary events $\overline{A}$ and $\overline{B}$ are also independent. Use the…
Prove analytically that if two events $A$ and $B$ are independent, then their complementary events $\overline{A}$ and $\overline{B}$ are also independent. Use the algebraic properties of probability and the fundamental definition of independence for this.