Events
A doctor interprets a diagnostic test, a company checks a batch of parts, a meteorologist combines "it rains" with "it is windy": in all these cases you have to reason with events and combine them with one another. Before calculating any probability, you need to be able to describe precisely what you are talking about.
Random experiments and the sample space
Classify the following experiments as deterministic or random: a) Measuring the time it takes for an object to fall from $10$ metres in a vacuum. b) Recording the colour…
Classify the following experiments as deterministic or random:
- a) Measuring the time it takes for an object to fall from $10$ metres in a vacuum.
- b) Recording the colour of the next car that passes along the street.
- c) Calculating the interest generated by a fixed-term bank deposit knowing the principal and the interest rate.
- d) Drawing a ball from a lottery drum.
Write down the sample space corresponding to the random experiment consisting of flipping a one-euro coin and a two-euro coin simultaneously, recording whether heads…
Write down the sample space corresponding to the random experiment consisting of flipping a one-euro coin and a two-euro coin simultaneously, recording whether heads ($C$) or tails ($X$) is obtained on each.
Events: certain, possible and impossible
An event is any subset of the sample space: a single outcome or a combination of several.
| Event | What it means | Example (urn with $5$ red and $3$ green balls) |
|---|---|---|
| Certain | It always happens | "drawing a red or green ball" |
| Possible | It may happen, but it is not certain | "drawing a red ball" |
| Impossible | It can never happen | "drawing a blue ball" |
In an urn there are $5$ red balls and $3$ green balls. If we draw a ball at random and observe its colour, invent a certain event and an impossible event.
In an urn there are $5$ red balls and $3$ green balls. If we draw a ball at random and observe its colour, invent a certain event and an impossible event.
When drawing a card from a Spanish deck of $40$ cards, we consider event $C = \{\text{drawing a face card}\}$. Describe in words what the complementary event $\bar{C}$…
When drawing a card from a Spanish deck of $40$ cards, we consider event $C = \{\text{drawing a face card}\}$. Describe in words what the complementary event $\bar{C}$ means and how many elements of the sample space make it up.
Operations with events
Just as with sets, events are combined using union, intersection, complement and difference.
In a market study on vehicle preferences, events $A = \{\text{el cliente prefiere coche eléctrico}\}$ and $B = \{\text{el cliente prefiere coche automático}\}$ are…
In a market study on vehicle preferences, events $A = \{\text{el cliente prefiere coche eléctrico}\}$ and $B = \{\text{el cliente prefiere coche automático}\}$ are defined. Express in words the meaning of the following events:
- a) $A \cap \bar{B}$
- b) $\overline{A \cup B}$
At a weather station, events $L = \{\text{it is raining today}\}$ and $V = \{\text{it is windy today}\}$ are defined. Write the following situations using mathematical…
At a weather station, events $L = \{\text{it is raining today}\}$ and $V = \{\text{it is windy today}\}$ are defined. Write the following situations using mathematical operations with events (union, intersection, complement):
- a) Today it is raining but it is not windy.
- b) Today it is neither raining nor windy.
- c) Either it is raining or it is windy, but not both at the same time.
Compatible and incompatible events
A die with $8$ faces numbered from $1$ to $8$ is rolled. State whether the following pairs of events are compatible or incompatible: a)…
A die with $8$ faces numbered from $1$ to $8$ is rolled. State whether the following pairs of events are compatible or incompatible:
- a) $A = \{\text{drawing an even number}\}$ and $B = \{\text{drawing a multiple of } 3\}$
- b) $C = \{\text{drawing a number greater than } 6\}$ and $D = \{\text{drawing a number less than } 5\}$
A traditional cubic die is rolled. We consider events $A = \{1, 2, 3\}$, $B = \{2, 4, 6\}$ and $C = \{4, 5, 6\}$. Find the elements that form the following events: a)…
A traditional cubic die is rolled. We consider events $A = \{1, 2, 3\}$, $B = \{2, 4, 6\}$ and $C = \{4, 5, 6\}$.
Find the elements that form the following events:
- a) $(A \cup B) \cap C$
- b) $\overline{A \cap B}$
Back to the start: If $A$ is "it rains" and $B$ is "it is windy", both happening at the same time is described by the intersection $A \cap B$, and at least one of the two happening by the union $A \cup B$.