Simple events and probability
Before a game, the forecast says "70% chance of rain in Guadalajara." In a school raffle, how likely is it that you win? To answer in an organized way, you have to do two things: describe the experiment well and count its outcomes well. That is what this page is about.
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 $5$.
Three fair coins are flipped simultaneously. Describe the sample space and calculate the probability of obtaining exactly two heads.
Three fair coins are flipped simultaneously. Describe the sample space and calculate the probability of obtaining exactly two heads.
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 |
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 $10$"
Probability: frequency and Laplace's rule
How do you assign a number to "how likely it is"? There are two ways.
Way 1: frequency. Repeat the experiment many times and calculate the relative frequency: $\dfrac{\text{times it occurred}}{\text{repetitions}}$. If the basketball player Diego makes $64$ of $80$ free throws, we estimate that he scores with probability $\dfrac{64}{80}=0.8$. As the number of repetitions increases (in a simulation with dice, for example), the relative frequency gets closer and closer to the theoretical probability.
Way 2: counting cases. If all the outcomes are equally likely (equally possible), there is no need to repeat anything.
Laplace's rule also works when there are many cases and it helps to count them in an organized way. With two dice there are $6\times 6=36$ equally likely outcomes; a sum of $8$ comes from $(2, 6),(3, 5),(4, 4),(5, 3),(6, 2)$, that is, $5$ cases, and $P=\dfrac{5}{36}$.
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 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?
In an urn there are $x$ red balls and $6$ white balls. Knowing that the probability of drawing a red ball at random is $\frac{2}{5}$, set up an equation and determine…
In an urn there are $x$ red balls and $6$ white balls. Knowing that the probability of drawing a red ball at random is $\frac{2}{5}$, set up an equation and determine the total number of balls in the urn.
Geometric probability
Sometimes the outcomes can't be counted: a dart can land on infinitely many points of a dartboard. If every point is equally likely, the probability depends on the size of the favorable region:
A circular target is made of three concentric circles with radii $10\text{ cm}$, $20\text{ cm}$ and $30\text{ cm}$. If a dart hits any point on the target at random…
A circular target is made of three concentric circles with radii $10\text{ cm}$, $20\text{ cm}$ and $30\text{ cm}$. If a dart hits any point on the target at random (guaranteeing it always lands inside the outer circle), calculate the probability that it lands in the middle annulus (the area between the radius of $10\text{ cm}$ and that of $20\text{ cm}$).
Probability of the union
Two events are mutually exclusive if they cannot occur at the same time ($A\cap B=\emptyset$). If they can coincide, they are non-mutually exclusive.
"A but not B": the difference event
Sometimes the problem asks for $A$ to occur and $B$ not to occur, that is, $A\cap B^c$. These are the outcomes of $A$ that fall outside the part shared with $B$.
If the problem doesn't give you the intersection, first calculate it with the union formula (as in the swimming and tennis example) and then subtract.
Consider the experiment of drawing a card from a Spanish deck of $40$ cards. State whether the events $A = $ "drawing a face card" and $B = $ "drawing a coin" are…
Consider the experiment of drawing a card from a Spanish deck of $40$ cards. State whether the events $A = $ "drawing a face card" and $B = $ "drawing a coin" are compatible or incompatible. Justify your answer.
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, $P(\overline{A \cap B})$.
A survey of $200$ coffee shop customers reveals that $120$ have coffee, $90$ have toast, and $40$ have neither. If a customer is chosen at random, calculate the…
A survey of $200$ coffee shop customers reveals that $120$ have coffee, $90$ have toast, and $40$ have neither. If a customer is chosen at random, calculate the probability that the customer has coffee but does NOT have toast.
Back to the beginning: the "70% chance of rain" is a probability estimated with frequencies: of days with similar conditions, it rained on $70$ out of every $100$. In the raffle, if there are $200$ tickets and you buy $1$, your probability is $\dfrac{1}{200}=0.5\,\%$, very far from $\frac12$.