Compound Events
Marcus makes $75\%$ of his free throws. In tonight's game he gets two free throws. How likely is he to make both? And how likely is he to make at least one? Tossing two coins, drawing two cards, or picking a bag and then a marble all work the same way: the experiment happens in stages, and each stage adds a probability to combine. By the end of this page you will answer Marcus's question both ways.
Multi-stage experiments
A tree diagram lists every outcome. Each branch carries its probability. Here are two coin tosses:
To get the probability at the end of a path, multiply along the branches: $\tfrac12 \times \tfrac12 = \tfrac14$. Why? Half of all the games start with heads, and then half of those halves end with heads again. You take a fraction of a fraction.
When one stage does not change the other (a coin and a die, two separate spinners), the stages are independent and you simply multiply the probability of each stage.
We toss two coins independently. Calculate the probability of obtaining exactly two tails.
We toss two coins independently. Calculate the probability of obtaining exactly two tails.
We roll a fair cubic die (with faces numbered from $1$ to $6$) and a coin. What is the probability of obtaining an even number on the die and heads on the coin?
We roll a fair cubic die (with faces numbered from $1$ to $6$) and a coin. What is the probability of obtaining an even number on the die and heads on the coin?
We spin the two independent spinners shown in the figure. Calculate the probability that the first spinner lands on the number $3$ and the second spinner lands on the…
We spin the two independent spinners shown in the figure. Calculate the probability that the first spinner lands on the number $3$ and the second spinner lands on the color green (V).
The multiplication rule
Sometimes the second stage depends on what happened in the first. Then you use the probability of the second stage given the first.
Using a table
When you are given counts, you can often read the answer straight from a two-way table.
In a restaurant, $60\%$ of the customers order the set menu and $40\%$ order à la carte. Of those who order the set menu, $70\%$ have dessert. Of those who order à la…
In a restaurant, $60\%$ of the customers order the set menu and $40\%$ order à la carte. Of those who order the set menu, $70\%$ have dessert. Of those who order à la carte, $50\%$ have dessert. Calculate the probability that a customer chosen at random orders the set menu and has dessert.
In a 10th grade class there are $12$ boys and $18$ girls. It is known that $4$ boys and $6$ girls wear glasses. If we choose a student at random, what is the probability…
In a 10th grade class there are $12$ boys and $18$ girls. It is known that $4$ boys and $6$ girls wear glasses. If we choose a student at random, what is the probability that it is a girl who wears glasses?
A sports center offers tennis and padel classes. There are $120$ members enrolled in these classes, and each member practices only one sport. It is known that $70$…
A sports center offers tennis and padel classes. There are $120$ members enrolled in these classes, and each member practices only one sport. It is known that $70$ members play padel and that there are $50$ women in total. Furthermore, $30$ women play tennis. If a member is chosen at random, calculate the probability that it is a man who plays padel.
With or without replacement
When you draw twice from the same bag, it matters whether the first item goes back.
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With replacement: the bag is the same each time, so every draw has the same probabilities. The draws are independent.
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Without replacement: the bag changes. The first item is gone, so both the favorable count and the total go down.
The same idea works for three or more draws. Each new factor has one fewer item on top and one fewer on the bottom.
In a box there are $5$ red balls and $3$ blue balls. A ball is extracted, its color is recorded, it is returned to the box and a second ball is extracted. Find the…
In a box there are $5$ red balls and $3$ blue balls. A ball is extracted, its color is recorded, it is returned to the box and a second ball is extracted. Find the probability that both balls are red.
Two cards are drawn consecutively without replacement from a Spanish deck of $40$ cards. Calculate the probability that both cards are Kings.
Two cards are drawn consecutively without replacement from a Spanish deck of $40$ cards. Calculate the probability that both cards are Kings.
Three cards are drawn consecutively and without replacement from a Spanish deck of $40$ cards. Calculate the probability of obtaining three cards of the coin suit.
Three cards are drawn consecutively and without replacement from a Spanish deck of $40$ cards. Calculate the probability of obtaining three cards of the coin suit.
Stopping rules and unknown counts
Drawing until something happens
If you draw until the first success, with a limit on the number of draws, each possible stopping point is its own path. Add the paths.
Finding how many objects there are
If the probability is given and the number of objects is unknown, call it $x$ and write the product as an equation.
An urn contains $x$ white balls and $5$ black balls. Two balls are extracted consecutively and without replacement. If we know that the probability that both balls are…
An urn contains $x$ white balls and $5$ black balls. Two balls are extracted consecutively and without replacement. If we know that the probability that both balls are white is $\frac{1}{6}$, set up a quadratic equation and determine the initial number of white balls $x$ that were in the urn.
Cards are drawn from a Spanish deck of $40$ cards one by one and without replacement, until the first Ace appears. The experiment ends the moment an Ace is drawn or when…
Cards are drawn from a Spanish deck of $40$ cards one by one and without replacement, until the first Ace appears. The experiment ends the moment an Ace is drawn or when a maximum of $3$ cards have been drawn. Calculate the probability that the experiment ends having found an Ace.
Two paths to the same result
Sometimes the first stage has several options, and each option leads to the result you want in a different way. Compute each path with the multiplication rule. Then add the paths.
We have two urns. Urn $A$ contains $4$ white balls and $2$ black balls. Urn $B$ contains $3$ white balls and $5$ black balls. We toss a fair coin; if it lands on heads,…
We have two urns. Urn $A$ contains $4$ white balls and $2$ black balls. Urn $B$ contains $3$ white balls and $5$ black balls. We toss a fair coin; if it lands on heads, we extract a ball from urn $A$, and if it lands on tails, we extract a ball from urn $B$. Find the probability that the extracted ball is white.
An electronic components factory has two machines. Machine $M_1$ produces $60\%$ of the components and machine $M_2$ the remaining $40\%$. It is known that $2\%$ of the…
An electronic components factory has two machines. Machine $M_1$ produces $60\%$ of the components and machine $M_2$ the remaining $40\%$. It is known that $2\%$ of the components produced by $M_1$ are defective, while $5\%$ of those produced by $M_2$ are defective. If a component is selected at random in the warehouse, what is the probability that it is defective?
A medical test to detect a disease has the following characteristics: if a person is ill, the test is positive in $95\%$ of cases. If the person is healthy, the test is…
A medical test to detect a disease has the following characteristics: if a person is ill, the test is positive in $95\%$ of cases. If the person is healthy, the test is positive (false positive) in $3\%$ of cases. It is known that $2\%$ of the population suffers from this disease. If a person is chosen at random and tested, what is the probability that the result is positive?
When the first stage changes the second
The first stage can decide what the second stage even looks like. Draw the branches, find the probability of the result on each, and add. A branch that can't produce the result contributes $0$, so you can leave it out.
Moving an object between containers
Here the first draw changes the contents of the second container. Split by what was moved.
In a game, a participant first draws a card from a Spanish deck of $40$ cards. If the drawn card is from the cup suit, they roll a fair cubic die (from $1$ to $6$). If…
In a game, a participant first draws a card from a Spanish deck of $40$ cards. If the drawn card is from the cup suit, they roll a fair cubic die (from $1$ to $6$). If the card is NOT from the cup suit, they toss a coin. What is the probability that the participant ends up obtaining a $5$ on the die?
We have two boxes. Box 1 contains $3$ red tokens and $2$ green tokens. Box 2 contains $1$ red token and $4$ green tokens. A token is extracted at random from box 1 and,…
We have two boxes. Box 1 contains $3$ red tokens and $2$ green tokens. Box 2 contains $1$ red token and $4$ green tokens. A token is extracted at random from box 1 and, without looking at its color, it is placed into box 2. Then, a token is extracted at random from box 2. Calculate the probability that this second token is red.
We have two urns. Urn $U_1$ contains $2$ red balls and $2$ black balls. Urn $U_2$ is initially empty. Two balls are drawn simultaneously from $U_1$ and placed into…
We have two urns. Urn $U_1$ contains $2$ red balls and $2$ black balls. Urn $U_2$ is initially empty. Two balls are drawn simultaneously from $U_1$ and placed into $U_2$. Then, a ball is drawn at random from $U_2$. Calculate the probability that the ball drawn from $U_2$ is red.
"At least one" and the complement
Back to Marcus. He makes $75\%$ of his free throws, and the two throws are independent.
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Both: $0.75 \times 0.75 = 0.5625$, about $56\%$.
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At least one: $1 - (0.25)(0.25) = 1 - 0.0625 = 0.9375$, about $94\%$.
Or: parallel and series
"A or B" is an "at least one" question, so use the complement. "A and B" is a plain product. In a pipe or circuit, a branch with parts in series needs all of them to work (multiply). Branches in parallel need at least one branch to work (complement).
Two basketball players, Emma and Chloe, each take a free throw. The probability that Emma scores is $0,8$ and the probability that Chloe scores is $0,7$. Their throws…
Two basketball players, Emma and Chloe, each take a free throw. The probability that Emma scores is $0,8$ and the probability that Chloe scores is $0,7$. Their throws are independent events. Calculate the probability that at least one of the two scores her free throw.
An archer has a probability of $0,85$ of hitting the center of the target on each shot. If he takes three consecutive and independent shots, find the probability that he…
An archer has a probability of $0,85$ of hitting the center of the target on each shot. If he takes three consecutive and independent shots, find the probability that he misses all three shots.
The following diagram shows a pipe system with three valves $A$, $B$ and $C$. The water flows from left to right. For the water to reach the end, it must pass through…
The following diagram shows a pipe system with three valves $A$, $B$ and $C$. The water flows from left to right. For the water to reach the end, it must pass through valve $A$ or through the lower branch containing valves $B$ and $C$ in series. The probabilities that the valves are open are $P(A) = 0,7$, $P(B) = 0,8$ and $P(C) = 0,9$. Knowing that the operation of each valve is independent, calculate the probability that the water manages to flow through the system.
Counting the right paths
Some questions have several paths that all count, such as "exactly one," or a series that ends when someone has won enough games. List every path, multiply along each, and add.
Areas as probabilities
When a dart hits a target at random, the probability of a region is its area divided by the total area.
Two esports teams, Alpha and Beta, play the final of a best-of-three tournament (the first to win two games is crowned champion). The probability that Alpha wins a game…
Two esports teams, Alpha and Beta, play the final of a best-of-three tournament (the first to win two games is crowned champion). The probability that Alpha wins a game is $0,6$ and that Beta wins is $0,4$ (there are no draws). The results of the games are independent. Calculate the probability that team Alpha wins the tournament.
At an international conference, $40\%$ of the attendees are European and $60\%$ are American. It is known that $30\%$ of the Europeans and $20\%$ of the Americans are…
At an international conference, $40\%$ of the attendees are European and $60\%$ are American. It is known that $30\%$ of the Europeans and $20\%$ of the Americans are bilingual. If we select two attendees at random with replacement, what is the probability that exactly one of the two is bilingual?
A dart is thrown at the target in the figure, formed by two concentric circles of radii $R$ and $2R$. The dart always hits the target and the probability of hitting a…
A dart is thrown at the target in the figure, formed by two concentric circles of radii $R$ and $2R$. The dart always hits the target and the probability of hitting a region is proportional to its area. If two darts are thrown independently, calculate the probability that one dart lands in the central zone (A) and the other in the outer zone (B).