Simple events
A quality engineer estimates what fraction of a batch will be defective, a meteorologist gives the percentage probability of rain, a casino calculates its margin on a roulette wheel: everything starts from the same place, describing the experiment precisely and counting its outcomes correctly. In Bachillerato, probability is formalised using set theory; this section reviews that foundation with the rigour you will need there.
Random experiment and sample space
Write down the sample space associated with the random experiment of drawing a ball from an urn containing $5$ balls numbered from $1$ to $5$.
Write down the sample space associated with the random experiment of drawing a ball from an urn containing $5$ balls numbered from $1$ to $5$.
In the experiment of rolling a $6$-sided die, classify the following events as elementary, compound, certain or impossible: $A = $ "rolling a $4$" $B = $ "rolling a…
In the experiment of rolling a $6$-sided die, classify the following events as elementary, compound, certain or impossible:
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$A = $ "rolling a $4$"
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$B = $ "rolling a number greater than $6$"
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$C = $ "rolling an even number"
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$D = $ "rolling a number less than $10$"
Types of events
| Event | Meaning |
|---|---|
| Certain | $A = E$: always occurs, $P(A) = 1$ |
| Impossible | $A = \emptyset$: never occurs, $P(A) = 0$ |
| Elementary | a single outcome |
| Compound | several outcomes at once |
A fair six-sided die with faces numbered from $1$ to $6$ is rolled. Determine the outcomes that form the complement of the event "rolling a number less than $3$" and…
A fair six-sided die with faces numbered from $1$ to $6$ is rolled. Determine the outcomes that form the complement of the event "rolling a number less than $3$" and calculate its probability.
Laplace's rule and the complementary event
A card is drawn at random from a Spanish deck of $40$ cards. Calculate the probability of obtaining a knight.
A card is drawn at random from a Spanish deck of $40$ cards. Calculate the probability of obtaining a knight.
A basketball player has taken $80$ free throws during the season, scoring $64$ of them. Using relative frequency as an estimate of probability, what is the probability…
A basketball player has taken $80$ free throws during the season, scoring $64$ of them. Using relative frequency as an estimate of probability, what is the probability that he misses his next free throw?
Probability of the union
Given that the probability of event $A$ occurring is $P(A) = \frac{9}{20}$, the probability of event $B$ occurring is $P(B) = \frac{3}{10}$, and the probability of both…
Given that the probability of event $A$ occurring is $P(A) = \frac{9}{20}$, the probability of event $B$ occurring is $P(B) = \frac{3}{10}$, and the probability of both occurring at the same time is $P(A \cap B) = \frac{1}{10}$, calculate the probability that at least one of the two events occurs.
In a class of $30$ students, $18$ do swimming, $15$ play tennis, and $5$ do neither sport. If a student is chosen at random, calculate the probability that they do both…
In a class of $30$ students, $18$ do swimming, $15$ play tennis, and $5$ do neither sport. If a student is chosen at random, calculate the probability that they do both sports simultaneously.
Let $A$ and $B$ be two events in a random experiment such that $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{3}$ and $P(A \cup B) = \frac{3}{4}$. Using the properties of…
Let $A$ and $B$ be two events in a random experiment such that $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{3}$ and $P(A \cup B) = \frac{3}{4}$. Using the properties of probability, calculate the probability of the complement of the intersection, that is, $P(\overline{A \cap B})$.
Back to the start: On a European roulette wheel with $37$ pockets ($0$ to $36$), betting on red gives $18$ favourable outcomes out of $37$: $P(\text{red}) = \dfrac{18}{37} \approx 0{,}4865$; the casino's margin arises because the probability that the $0$ comes up (so that both red and black lose) is $\dfrac{1}{37} \approx 2{,}7\,\%$.