Compound Events
You toss two coins, draw two cards, or first pick an urn and then a ball: these are compound experiments, made up of several stages. How do you combine the probabilities of each stage? For example, what is the probability of drawing a white ball if a coin decides which urn it is drawn from? We'll solve it at the end of the page.
Multi-stage experiments
If the stages do not influence each other (two coins, a die and a coin, two spinners), the probability of each stage is the usual one and you just multiply. For example, for two tails: $\frac12 \times \frac12 = \frac14$.
Check the calculation with a simulation
How do you know that $\frac14$ is reasonable? Simulate it. With the chance tool above, or in a spreadsheet, repeat the experiment many times: in a cell type =RANDBETWEEN(0,1) for a coin, use 0 = tails and 1 = heads, and copy the formula into two columns, one per coin. Mark each row where both coins are tails and calculate the relative frequency:
With $20$ repetitions the result can end up far from $0.25$. With $1{,}000$ it gets very close: as the number of repetitions increases, the relative frequency tends toward the theoretical probability. Do this with each exercise in this section whenever you doubt your result.
We flip two coins independently. Calculate the probability of obtaining exactly two tails.
We flip two coins independently. Calculate the probability of obtaining exactly two tails.
We roll a fair six-sided die (with faces numbered from $1$ to $6$) and flip a coin. What is the probability of obtaining an even number on the die and a head on the coin?
We roll a fair six-sided die (with faces numbered from $1$ to $6$) and flip a coin. What is the probability of obtaining an even number on the die and a head 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 green…
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 green (V).
On a street there are two independent traffic lights. The probability of the first one being green is $0.4$ and the probability of the second one being green is $0.5$.…
On a street there are two independent traffic lights. The probability of the first one being green is $0.4$ and the probability of the second one being green is $0.5$. Find the probability of encountering both traffic lights on green when travelling down that street.
The product rule
The same idea solves problems where the unknown is the number of balls.
In a box there are $5$ red balls and $3$ blue balls. A ball is drawn, its color recorded, replaced in the box, and a second ball is drawn. Find the probability that both…
In a box there are $5$ red balls and $3$ blue balls. A ball is drawn, its color recorded, replaced in the box, and a second ball is drawn. Find the probability that both balls are red.
From a Spanish deck of $40$ cards, two cards are drawn consecutively without replacement. Calculate the probability that both cards are Kings.
From a Spanish deck of $40$ cards, two cards are drawn consecutively without replacement. Calculate the probability that both cards are Kings.
In a restaurant, $60\%$ of customers order the set menu and $40\%$ order à la carte. Of those ordering the set menu, $70\%$ have dessert. Of those ordering à la carte,…
In a restaurant, $60\%$ of customers order the set menu and $40\%$ order à la carte. Of those ordering the set menu, $70\%$ have dessert. Of those ordering à la carte, $50\%$ have dessert. Calculate the probability that a customer chosen at random orders the set menu and has dessert.
In a Year 11 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 Year 11 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 they are a girl who wears glasses?
Three cards are drawn consecutively and without replacement from a Spanish deck of $40$ cards. Calculate the probability of obtaining three cards of the suit of coins.
Three cards are drawn consecutively and without replacement from a Spanish deck of $40$ cards. Calculate the probability of obtaining three cards of the suit of coins.
At a sports center, tennis and padel classes are offered. There are $120$ members enrolled in these classes, and each member plays only one sport. It is known that $70$…
At a sports center, tennis and padel classes are offered. There are $120$ members enrolled in these classes, and each member plays 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 they are a man who plays padel.
An urn contains $x$ white balls and $5$ black balls. Two balls are drawn consecutively without replacement. Knowing that the probability of both balls being white is…
An urn contains $x$ white balls and $5$ black balls. Two balls are drawn consecutively without replacement. Knowing that the probability of both balls being white is $\frac{1}{6}$, set up a quadratic equation and determine the initial number of white balls $x$ that were in the urn.
Two paths to the same result
When the first stage has several options and each one leads to a different second stage, you have to add the probability of all the paths that lead to the desired result. This is called total probability: you calculate $P(\text{result})$ even if you don't know which path it came through.
That closes the opening question: the probability of white is $\frac{25}{48}\approx 0.52$.
That is the question the exercises in this section ask: the probability of a positive result. But you can also ask the reverse: if someone tested positive, how likely is it that they are sick? Imagine $1{,}000$ people: $45$ sick with a positive result and $38$ healthy with a positive result, that is, $83$ positives in total. Of those $83$, $45$ are sick, so the probability is $\frac{45}{83}\approx 0.54$. It is the probability of the path "sick and positive" divided by the total probability of a positive: $\frac{0.045}{0.083}$. This idea (Bayes) is studied in detail in the sections on dependence and on trees.
The same thing works when the first stage is a draw and the result is not seen: you separate the possible cases of the first stage, calculate each path, and add them up.
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 flip 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 flip a fair coin; if it lands on heads, we draw a ball from urn $A$, and if it lands on tails, we draw a ball from urn $B$. Find the probability that the drawn ball is white.
An electronic component 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 component 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 from the warehouse, what is the probability that it is defective?
In a game, a participant first draws a card from a Spanish deck of $40$ cards. If the drawn card is of the suit of cups, he rolls a fair six-sided die (from $1$ to $6$).…
In a game, a participant first draws a card from a Spanish deck of $40$ cards. If the drawn card is of the suit of cups, he rolls a fair six-sided die (from $1$ to $6$). If the card is NOT of cups, he flips a coin. What is the probability that the participant ends up getting a $5$ on the die?
We have two boxes. Box 1 contains $3$ red counters and $2$ green counters. Box 2 contains $1$ red counter and $4$ green counters. A counter is drawn at random from Box 1…
We have two boxes. Box 1 contains $3$ red counters and $2$ green counters. Box 2 contains $1$ red counter and $4$ green counters. A counter is drawn at random from Box 1 and, without looking at its color, is placed into Box 2. Next, a counter is drawn at random from Box 2. Calculate the probability that this second counter is red.
A medical test to detect a disease has the following characteristics: if a person is ill, the test gives a positive result in $95\%$ of cases. If the person is healthy,…
A medical test to detect a disease has the following characteristics: if a person is ill, the test gives a positive result in $95\%$ of cases. If the person is healthy, the test gives a positive result (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?
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 $U_2$. Next, a ball is drawn at random from $U_2$. Calculate the probability that the ball drawn from $U_2$ is red.
Adding up the possible orders
Sometimes the result can be achieved in several ways depending on the order in which things happen. Each order is a different path: you multiply along each one and then add the paths.
If an event has probability $p$ on each independent attempt, "exactly one out of two" has probability $p(1-p)+(1-p)p=2p(1-p)$: there are two orders.
Two e-sports teams, Alpha and Beta, play the final of a best-of-three tournament (the first to win two games becomes champion). The probability of Alpha winning a game…
Two e-sports teams, Alpha and Beta, play the final of a best-of-three tournament (the first to win two games becomes champion). The probability of Alpha winning a game is $0.6$ and that of Beta winning is $0.4$ (there are no draws). The game results 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?
Probability as a ratio of areas
On a dartboard or a spinner, the point of impact can land anywhere on the surface. If all points are equally likely, the probability of landing in a region is the fraction of the total area that it occupies:
$$P(\text{region})=\frac{\text{area of the region}}{\text{total area}}$$It is the same idea as "favorable cases over possible cases," but measuring surfaces.
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 region (A) and the other in the outer region (B).
"At least one"
The complement also works when the attempts are dependent: you multiply conditional probabilities.
Two basketball players, Marta and Laura, each take a free throw. The probability that Marta scores is $0.8$ and the probability that Laura scores is $0.7$. Their shots…
Two basketball players, Marta and Laura, each take a free throw. The probability that Marta scores is $0.8$ and the probability that Laura scores is $0.7$. Their shots are independent events. Calculate the probability that at least one of them scores her free throw.
An archer has a probability of $0.85$ of hitting the bullseye on each shot. If he takes three consecutive independent shots, find the probability that he misses all…
An archer has a probability of $0.85$ of hitting the bullseye on each shot. If he takes three consecutive independent shots, find the probability that he misses all three shots.
The following diagram shows a pipe system with three valves $A$, $B$ and $C$. Water flows from left to right. For water to reach the end, it must pass through valve $A$…
The following diagram shows a pipe system with three valves $A$, $B$ and $C$. Water flows from left to right. For 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 of the valves being 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 water successfully passes through the system.
Cards are drawn one by one without replacement from a Spanish deck of $40$ cards until the first Ace appears. The experiment ends as soon as an Ace is drawn or when a…
Cards are drawn one by one without replacement from a Spanish deck of $40$ cards until the first Ace appears. The experiment ends as soon as 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.