Simple events
Before a Leaving Cert maths class, Aoife and Conor bet on whether the Luas will arrive on time, Met Éireann gives a 40% chance of rain, and a Lotto draw picks six numbers. Everything starts from the same place: describing the experiment precisely and counting its outcomes correctly. In Leaving Certificate Maths you use sets and Venn diagrams to do this, and this section builds that foundation.
Random experiment and sample space
Two coins give the sample space $\{HH, HT, TH, TT\}$: write the outcomes in order so that all of them are equally likely.
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 $10$"
Two fair six-sided dice are rolled simultaneously, and the sum of the scores on the top faces is recorded. Calculate the probability that the sum is exactly $8$.
Two fair six-sided dice are rolled simultaneously, and the sum of the scores on the top faces is recorded. Calculate the probability that the sum is exactly $8$.
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.
Equally likely outcomes and the complementary event
In a toolbox there are $12$ screws, $8$ nuts and $5$ nails. If an item is picked at random, calculate the probability that it is a nut.
In a toolbox there are $12$ screws, $8$ nuts and $5$ nails. If an item is picked at random, calculate the probability that it is a nut.
A card is drawn at random from a deck of $40$ cards that includes knights. Calculate the probability of obtaining a knight.
A card is drawn at random from a deck of $40$ cards that includes knights. 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
Smallest possible overlap
The union can never be bigger than the whole sample space: $P(A \cup B) \le 1$. Since $P(A \cap B) = P(A) + P(B) - P(A \cup B)$, this gives $P(A \cap B) \ge P(A) + P(B) - 1$. The overlap can never be negative either, so the smallest possible value of $P(A \cap B)$ is the larger of $0$ and $P(A) + P(B) - 1$.
Given the sample space $E = \{1, 2, 3, 4, 5, 6, 7, 8\}$ and the events $A = \{2, 4, 6, 8\}$ and $B = \{1, 2, 3, 4\}$, determine the outcomes that form the events…
Given the sample space $E = \{1, 2, 3, 4, 5, 6, 7, 8\}$ and the events $A = \{2, 4, 6, 8\}$ and $B = \{1, 2, 3, 4\}$, determine the outcomes that form the events $A \cup B$ and $A \cap B$.
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: A fair die is rolled in a game at a school fair. The probability of an even number is $\dfrac{3}{6} = \dfrac{1}{2}$, and the probability of NOT getting a $6$ is $1 - \dfrac{1}{6} = \dfrac{5}{6}$. Counting outcomes carefully is what makes these calculations reliable.