Laplace's rule, $P(A) = \frac{\text{favorables}}{\text{posibles}}$, is easy to state but sometimes very hard to apply: how many hands of cards are there? How many ways can people be arranged in a row? Counting them one by one is impossible as soon as the numbers get big. This section revisits the factorial and the binomial coefficients that you already used with the binomial theorem and puts them to work inside Laplace's rule. The Combinatorics section that follows will give them their formal names (permutations, variations); here we already use them as a tool.
Before you start, make sure you know: Tree diagrams (4th ESO)
Tree diagrams and the multiplication principle
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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?
When several elements are chosen and arranged in order without repetition (a podium, an order of arrival, a set of distinct posts), the number of possible outcomes is a product of decreasing factors, related to the factorial you already know.
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In a $100$-metre 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 $100$-metre 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?
When several elements are chosen without regard to order (a committee, a sample, a hand of cards), the number of outcomes is calculated with the binomial coefficient $\binom{n}{k}$ that you already used in the binomial theorem.
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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 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?
Cases with restrictions: elements together or repeated
Some problems require certain elements to stay together, or involve counting anagrams with repeated letters: in both cases it is best to group the elements or divide by the repetitions before applying Laplace's rule.
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$5$ 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?
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?
Back to the start: To count the 5-card hands from a Spanish 40-card deck, order does not matter and there are $\binom{40}{5} = 658\,008$ possible hands, whereas in a row of $n$ people order does matter and there are $n!$ different arrangements.
Summary
Let's practice! 10 medium exercises in this section