Before you start, make sure you know: Introduction (High School Year 1 (semesters 1 and 2))
Before you start, make sure you know: Tree diagrams (High School Year 1 (semesters 1 and 2))
In the Melate lottery, $6$ numbers are chosen from a total of $56$. What is the probability of winning? Laplace's rule says $P(A)=\dfrac{\text{favorable cases}}{\text{possible cases}}$, but how do you count millions of combinations without writing them out one by one? This section gives you the tools to count without listing: the multiplication principle, arrangements, combinations, and a few tricks for the hard cases. At the end we come back to the Melate.
Multiplication Principle and Tree Diagrams
When a choice is made in stages, a tree diagram shows all the paths. Counting the paths means multiplying.
The reason: for each option in the first stage, all the branches of the second stage open up, and so on.
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A restaurant offers $4$ starters, $3$ main courses and $2$ desserts. If a customer chooses their menu at random, what is the probability that they choose exactly the…
A restaurant offers $4$ starters, $3$ main courses and $2$ desserts. If a customer chooses their menu at random, what is the probability that they choose exactly the salad for starter, the salmon for main course and the cake for dessert?
If you choose several different elements and the order changes the result (a podium, offices, finishing order), each choice reduces the options for the next one.
The first place can be filled in $n$ ways; the second in $n-1$ (one has already been used); the third in $n-2$, and so on.
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In a $100$-meter sprint, $6$ athletes take part. If all have the same probability of winning and there are no ties, what is the probability that the finishing order is…
In a $100$-meter sprint, $6$ athletes take part. If all have the same probability of winning and there are no ties, what is the probability that the finishing order is exactly A, B, C, D, E, F?
In a chess club with $10$ members, a president, a secretary and a treasurer are to be chosen at random. What is the probability that Ana is chosen as president, Carlos…
In a chess club with $10$ members, a president, a secretary and a treasurer are to be chosen at random. What is the probability that Ana is chosen as president, Carlos as secretary and Elena as treasurer?
All possible words, whether meaningful or not, are formed by rearranging the letters of the word "ROMA". If one of these words is chosen at random, what is the…
All possible words, whether meaningful or not, are formed by rearranging the letters of the word "ROMA". If one of these words is chosen at random, what is the probability that it starts with the letter "M"?
Four friends board a bus that has $6$ stops remaining until the end of the route. If each friend is equally likely to get off at any of the stops, what is the…
Four friends board a bus that has $6$ stops remaining until the end of the route. If each friend is equally likely to get off at any of the stops, what is the probability that all four friends get off at different stops?
If only who is chosen matters (a committee, a sample, a hand of cards), the order is irrelevant. Each group of $k$ elements appears $k!$ times when counting arrangements, so you divide by $k!$.
When there are two types of elements and you are asked how many of each type, multiply the combinations for each type.
It is also useful for choosing positions. When you flip a coin $6$ times, what is the probability of $2$ heads? There are $2^6 = 64$ possible sequences; the favorable ones are the ways to choose which $2$ flips come up heads: $\binom62 = 15$. So $P=\dfrac{15}{64}$.
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From a group of $8$ students, $3$ are chosen at random to form a representative committee. What is the probability that the chosen ones are exactly the $3$ oldest…
From a group of $8$ students, $3$ are chosen at random to form a representative committee. What is the probability that the chosen ones are exactly the $3$ oldest students in the group?
In a tech company, $7$ engineers and $5$ economists are employed. A project team of $4$ people is to be chosen at random. What is the probability that the team consists…
In a tech company, $7$ engineers and $5$ economists are employed. A project team of $4$ people is to be chosen at random. What is the probability that the team consists of exactly $2$ engineers and $2$ economists?
A Spanish deck has $40$ cards: $4$ suits (coins, cups, swords, and clubs) of $10$ cards each. To draw $3$ of the same suit, first choose the suit ($4$ ways) and then the $3$ cards within it, using combinations.
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In a box there are $5$ red balls and $4$ blue balls. $2$ balls are drawn at random simultaneously. What is the probability that both balls are red?
In a box there are $5$ red balls and $4$ blue balls. $2$ balls are drawn at random simultaneously. What is the probability that both balls are red?
From a Spanish deck of $40$ cards, $3$ cards are drawn at random without replacement. What is the probability that all three cards are of the same suit?
From a Spanish deck of $40$ cards, $3$ cards are drawn at random without replacement. What is the probability that all three cards are of the same suit?
When the problem requires certain elements to stay together, treat them as a block. When a position has a condition (for example, an even number), count the restricted position first.
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Numbers with $4$ distinct digits are formed at random using the digits $1, 2, 3, 4, 5, 6$. What is the probability that the number formed is even?
Numbers with $4$ distinct digits are formed at random using the digits $1, 2, 3, 4, 5, 6$. What is the probability that the number formed is even?
$5$ mathematics books and $3$ physics books are to be placed at random on a shelf. What is the probability that all the physics books are placed together?
$5$ mathematics books and $3$ physics books are to be placed at random on a shelf. What is the probability that all the physics books are placed together?
If some elements are identical, swapping them does not produce a new arrangement, so you divide by the arrangements of the repeated ones.
Circular table. At a round table, rotating everyone one seat over does not change the arrangement. To avoid counting rotations, fix one person and arrange the others: with $n$ people there are $(n-1)!$ arrangements.
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All possible words are formed by rearranging the letters of the word "BANANA". What is the probability that the three letters "A" appear together consecutively?
All possible words are formed by rearranging the letters of the word "BANANA". What is the probability that the three letters "A" appear together consecutively?
Eight people, including two siblings, sit at random around a circular table. What is the probability that the two siblings do not sit in adjacent seats?
Eight people, including two siblings, sit at random around a circular table. What is the probability that the two siblings do not sit in adjacent seats?
Complement. When a problem asks for "at least one," it is shorter to count the opposite case ("none") and subtract: $P(\text{at least one})=1-P(\text{none})$.
Reduced sample space. If the statement says "given that…," something has already happened. Count only what is left to happen.
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A batch of $20$ microchips contains $4$ defective ones. A random sample of $3$ microchips is selected for inspection. What is the probability that there is at least one…
A batch of $20$ microchips contains $4$ defective ones. A random sample of $3$ microchips is selected for inspection. What is the probability that there is at least one defective microchip in the sample?
A fair $6$-sided die is rolled $6$ consecutive times. What is the probability that all numbers from $1$ to $6$ appear exactly once, given that the first three rolls…
A fair $6$-sided die is rolled $6$ consecutive times. What is the probability that all numbers from $1$ to $6$ appear exactly once, given that the first three rolls resulted exclusively in even numbers?
To count a hand or a committee with a specific structure, divide it into stages (multiply) and, if there are several ways to meet the condition, add the cases.
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A multiple-choice exam consists of $10$ questions, each with $4$ answer options, only one of which is correct. If a student answers all questions at random, what is the…
A multiple-choice exam consists of $10$ questions, each with $4$ answer options, only one of which is correct. If a student answers all questions at random, what is the probability that they get exactly $6$ questions right?
From a standard deck of $52$ cards (consisting of $13$ different ranks across $4$ suits), $5$ cards are drawn at random. What is the probability of getting a full house…
From a standard deck of $52$ cards (consisting of $13$ different ranks across $4$ suits), $5$ cards are drawn at random. What is the probability of getting a full house (that is, three cards of one rank and two cards of another rank)?
A university department consists of $6$ tenured professors and $8$ associate professors. A committee of $5$ people is chosen at random. What is the probability that the…
A university department consists of $6$ tenured professors and $8$ associate professors. A committee of $5$ people is chosen at random. What is the probability that the committee has a majority of tenured professors (that is, $3$ or more)?
Three more situations that appear in hard problems.
Distributing objects into bins (or people onto floors). If each of $k$ objects goes to any of $m$ places, there are $m^k$ possible distributions. For a distribution with conditions, classify by the shape of the distribution and count each shape by stages.
Geometry with vertices. A triangle inscribed in a circle is a right triangle if and only if one of its sides is a diameter (Thales' theorem). Count the diameters and the third vertex.
Derangements. A derangement is an arrangement in which no element ends up in its original place. With $2$ objects there is $1$ derangement; with $3$ there are $2$.
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$3$ distinct vertices of a regular decagon are chosen at random. What is the probability that the triangle formed by these $3$ vertices is a right-angled triangle?
$3$ distinct vertices of a regular decagon are chosen at random. What is the probability that the triangle formed by these $3$ vertices is a right-angled triangle?
On the ground floor of a $8$-storey building, $5$ people get into the lift. If each person chooses at random the floor at which they get off (from $1$ to $8$), what is…
On the ground floor of a $8$-storey building, $5$ people get into the lift. If each person chooses at random the floor at which they get off (from $1$ to $8$), what is the probability that exactly two people get off on the same floor and the other three get off on floors that are different from each other and different from that of the pair?
Four people leave their coats in a theater cloakroom. Upon leaving, due to a power cut, the attendant hands back the four coats at random. What is the probability that…
Four people leave their coats in a theater cloakroom. Upon leaving, due to a power cut, the attendant hands back the four coats at random. What is the probability that exactly one person receives their own coat?
Back to the beginning: in the Melate the order of the numbers doesn't matter, so there are $\binom{56}{6}=32{,}468{,}436$ possible combinations. Hitting the one winning combination has probability $\dfrac{1}{32{,}468{,}436}$, roughly one in $32$ million.
Summary
Let's practice! 8 medium exercises in this section