Combinatorial numbers
To organise the graduation party, a committee of $3$ people has to be chosen from $20$ volunteers. To calculate the probability that a multiple-choice test has exactly $k$ correct answers by pure guesswork, we need to count in how many ways those $k$ correct answers can be placed among all the questions. Both problems are solved with the same tool: combinatorial numbers.
The factorial
Calculate and simplify the value of the following expressions without using a calculator, applying the properties of factorials: a) $\frac{8!}{6!}$ b) $\binom{7}{2}$ c)…
Calculate and simplify the value of the following expressions without using a calculator, applying the properties of factorials:
- a) $\frac{8!}{6!}$
- b) $\binom{7}{2}$
- c) $5! - 4!$
Combinatorial numbers: definition and meaning
To organize the graduation party for 2.º de Bachillerato, a party committee made up of $3$ students must be chosen. If there are $20$ volunteer students in the class, in…
To organize the graduation party for 2.º de Bachillerato, a party committee made up of $3$ students must be chosen. If there are $20$ volunteer students in the class, in how many different ways can this group be chosen? (The order in which the students are chosen does not matter: only who is on the committee counts.) Use the combinatorial number formula:
$$\binom{n}{r} = \frac{n!}{r!(n-r)!}$$Properties: symmetry and Stifel's rule
In the section on binomial coefficients, there is a symmetry property stating that $\binom{n}{r} = \binom{n}{n-r}$. Verify this property by calculating the values of…
In the section on binomial coefficients, there is a symmetry property stating that $\binom{n}{r} = \binom{n}{n-r}$.
Verify this property by calculating the values of $\binom{10}{8}$ and $\binom{10}{2}$. What conclusion do you reach regarding the computational effort when $r$ is a number close to $n$?
Prove, using the definition of binomial coefficient using factorials, the following identity (known as Pascal's rule or Stifel's identity): $$\binom{n}{k} +…
Prove, using the definition of binomial coefficient using factorials, the following identity (known as Pascal's rule or Stifel's identity):
$$\binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k}$$Apply this result to calculate the value of $\displaystyle \binom{10}{4} + \binom{10}{3}$ without calculating each term separately.
Discrete and continuous variables
Before applying combinatorics to probability, we need to identify what type of random variable is being counted.
Before studying the binomial distribution, it is essential to distinguish the type of variable. Classify the following random variables as discrete or continuous: 1. The…
Before studying the binomial distribution, it is essential to distinguish the type of variable. Classify the following random variables as discrete or continuous:
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The number of "likes" an Instagram post receives in one hour.
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The exact time it takes a student to get to school.
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The number of people who test positive in an antigen test in a sample of $50$ individuals.
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The height of the players in the Spanish national basketball team.
From combinatorics to binomial probability
When an experiment with two outcomes (success/failure) is repeated $n$ times, the combinatorial number is used to count in how many ways the $k$ successes can be placed among the $n$ trials: this count is the missing piece of the binomial formula.
A study states that $70\%$ of Spanish teenagers use TikTok daily. If we randomly select a group of $8$ teenagers from a secondary school: a) Identify the parameters $n$…
A study states that $70\%$ of Spanish teenagers use TikTok daily. If we randomly select a group of $8$ teenagers from a secondary school:
- a) Identify the parameters $n$ and $p$ of the binomial distribution that this situation follows.
- b) Define the random variable $X$ associated with this experiment.
- c) What is the probability that exactly $8$ teenagers use the app? (State only the formulation of the formula).
Given that for a binomial distribution $B(n, p)$, the probability of obtaining $k$ successes is: $$P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$$ Calculate the…
Given that for a binomial distribution $B(n, p)$, the probability of obtaining $k$ successes is:
$$P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$$Calculate the probability of obtaining exactly $2$ heads when tossing a fair coin $4$ times.
At a high-performance sports center, a group of 4 athletes is to be selected to represent Spain in an international relay competition. If there are 12 athletes available…
At a high-performance sports center, a group of 4 athletes is to be selected to represent Spain in an international relay competition. If there are 12 athletes available at the center (7 men and 5 women), calculate:
- a) The total number of different ways the team of 4 people can be chosen.
- b) In how many of those combinations would the team consist of exactly 2 men and 2 women?
- c) Use the properties of binomial coefficients to prove, without calculating the numerical value, that:
Back to the start: In a multiple-choice test of $n$ questions with $4$ options per question, the probability of getting exactly $k$ right by guessing is $P(X=k) = \binom{n}{k} \cdot 0{,}25^k \cdot 0{,}75^{n-k}$, where the combinatorial number counts all the possible ways of distributing the correct answers.