Laplace's Rule
A bag has 5 red marbles and 3 blue ones. Without looking, you reach in and pull one out. Which is more likely: red or blue? At a glance you may have a hunch, but with math we can give an exact number.
Favorable outcomes and possible outcomes
When all the outcomes of an experiment are equally likely (a fair die, the cards in a deck, marbles that are identical except for color), we use Laplace's rule:
$$P(A) = \frac{\text{número de casos favorables a } A}{\text{número de casos posibles}}$$Change the quantities and watch how the frequencies approach the calculated probability:
A fair die is rolled. Find the probability of each event: a) Rolling a $3$. b) Rolling an even number. c) Rolling a number greater than $4$. d) Not rolling a $6$.
A fair die is rolled. Find the probability of each event:
a) Rolling a $3$.
b) Rolling an even number.
c) Rolling a number greater than $4$.
d) Not rolling a $6$.
A ball is drawn at random from the bag shown in the figure, which contains $5$ red balls and $3$ green balls. Find the probability of: a) Drawing a red ball. b) Drawing…
A ball is drawn at random from the bag shown in the figure, which contains $5$ red balls and $3$ green balls.
Find the probability of:
a) Drawing a red ball.
b) Drawing a green ball.
c) Drawing a yellow ball.
Assign a probability to each event and explain why: a) Rolling a die and getting a number less than $7$. b) Rolling a die and getting a $8$. c) Drawing a card from a…
Assign a probability to each event and explain why:
a) Rolling a die and getting a number less than $7$.
b) Rolling a die and getting a $8$.
c) Drawing a card from a Spanish deck and getting coins, cups, swords, or clubs.
The complementary event
a) If the probability of drawing coins in a Spanish deck of cards is $\frac{1}{4}$, what is the probability of not drawing coins? b) If $P(A) = 0.35$, what is the…
a) If the probability of drawing coins in a Spanish deck of cards is $\frac{1}{4}$, what is the probability of not drawing coins?
b) If $P(A) = 0.35$, what is the probability of the complementary event of $A$?
c) An urn contains only red, blue, and green balls. If the probability of drawing a red ball is $\frac{2}{7}$ and that of drawing a blue ball is $\frac{3}{7}$, what is the probability of drawing a green ball?
In a bag there are $5$ red balls, $3$ blue balls and $2$ white balls. A ball is drawn at random. Calculate the probability that: a) It is not white. b) It is white or…
In a bag there are $5$ red balls, $3$ blue balls and $2$ white balls. A ball is drawn at random. Calculate the probability that:
a) It is not white.
b) It is white or blue.
c) It is yellow.
d) It is not yellow.
The chart shows the sweets in a bag. One sweet is drawn at random. Find the probability that it is: a) Strawberry. b) Lemon or mint. c) Not apple. d) Orange.
The chart shows the sweets in a bag. One sweet is drawn at random.
Find the probability that it is:
a) Strawberry.
b) Lemon or mint.
c) Not apple.
d) Orange.
Events with "or"
For "getting A or B," you count the favorable outcomes of either one. If an outcome meets both conditions, it is counted only once.
A number from $1$ to $50$ is chosen at random. Find the probability that it is: a) A multiple of $4$. b) A multiple of $5$. c) A multiple of $4$ and $5$. d) A multiple…
A number from $1$ to $50$ is chosen at random. Find the probability that it is:
a) A multiple of $4$.
b) A multiple of $5$.
c) A multiple of $4$ and $5$.
d) A multiple of $4$ or $5$.
A card is drawn from a Spanish deck of $40$ cards. Calculate the probability that it is: a) Coins or a face card. b) Cups or an ace. c) Neither a face card nor an ace.…
A card is drawn from a Spanish deck of $40$ cards. Calculate the probability that it is:
a) Coins or a face card.
b) Cups or an ace.
c) Neither a face card nor an ace.
d) Clubs or a number card less than $4$ (the ace counts as $1$).
An NGO sells $200$ raffle tickets numbered from $000$ to $199$ for a charity draw. One ticket is drawn at random. Calculate the probability that the winning number: a)…
An NGO sells $200$ raffle tickets numbered from $000$ to $199$ for a charity draw. One ticket is drawn at random. Calculate the probability that the winning number:
a) Contains the digit $7$.
b) Is a multiple of $10$.
c) Starts with the digit $1$.
d) Has all three digits equal.
Compound experiments
With two dice, two coins, or a coin and a die, you first count all the possible outcomes with a two-way table or a tree diagram, and then you apply Laplace's rule.
A coin is tossed and a die is rolled. Find the probability of getting: a) Heads and a $6$. b) Tails and an odd number. c) Heads and a number less than $5$.
A coin is tossed and a die is rolled. Find the probability of getting:
a) Heads and a $6$.
b) Tails and an odd number.
c) Heads and a number less than $5$.
Three coins are tossed. Using a tree diagram, find the probability of getting: a) Exactly two heads. b) At least one tail. c) Three identical outcomes. d) Exactly one…
Three coins are tossed. Using a tree diagram, find the probability of getting:
a) Exactly two heads.
b) At least one tail.
c) Three identical outcomes.
d) Exactly one head.
In a family with two children, Nerea reasons: 'There are three possible outcomes: two boys, two girls, or one of each; therefore, the probability that both are boys is…
In a family with two children, Nerea reasons: 'There are three possible outcomes: two boys, two girls, or one of each; therefore, the probability that both are boys is $\frac{1}{3}$.'
a) Explain why her reasoning is incorrect.
b) Write down the correct sample space, taking birth order into account, and calculate the correct probability.
c) What is the probability that there is at least one girl?
Spinners, clocks, and unequal sectors
If a spinner doesn't have equal sectors, each sector is no longer "one outcome." What is proportional to the probability is its angle.
The same reasoning works for a clock: if you spin the hour hand at random, the full turn is $12$ hours. For example, the probability that it points between $3$ and $4$ is $\dfrac{1}{12}$.
The spinner shown in the figure has three sectors with the indicated angles. Find the probability that the arrow stops on: a) The red sector. b) The green sector. c) A…
The spinner shown in the figure has three sectors with the indicated angles.
Find the probability that the arrow stops on:
a) The red sector.
b) The green sector.
c) A sector that is not yellow.
The spinner shown in the figure has four sectors with the indicated angles. Find the probability that the arrow stops on: a) The red sector. b) The yellow sector. c) The…
The spinner shown in the figure has four sectors with the indicated angles.
Find the probability that the arrow stops on:
a) The red sector.
b) The yellow sector.
c) The blue sector or the green sector.
d) A sector that is not red.
The hour hand of a clock is spun at random. Calculate the probability that it points to a time: a) Between $3$ o'clock and $4$ o'clock. b) Before $6$ o'clock. c) After…
The hour hand of a clock is spun at random. Calculate the probability that it points to a time:
a) Between $3$ o'clock and $4$ o'clock.
b) Before $6$ o'clock.
c) After $10$.
d) Between $2$ and $8$.
Data tables and changing outcomes
Sometimes the possible outcomes come from a table of data, or they get reduced because we already know something ("given that…," or "without replacement" of the card or ball).
When data is missing: set up an equation
The students at a school who do extracurricular sports are distributed as follows: | | soccer | swimming | tennis | |---|---|---|---| | $6.º$ Elementary School | $14$ |…
The students at a school who do extracurricular sports are distributed as follows:
| soccer | swimming | tennis | |
|---|---|---|---|
| $6.º$ Elementary School | $14$ | $7$ | $4$ |
| $1.º$-$2.º$ Middle School | $16$ | $4$ | $15$ |
A student is chosen at random. Calculate the probability that:
a) They are in elementary school.
b) They do swimming.
c) They are in elementary school and play tennis.
d) Given that they are in middle school, they play tennis.
In a class of $30$ students there are $18$ girls, of whom $12$ have passed all subjects. In total, $10$ students in the class have failed at least one subject. a)…
In a class of $30$ students there are $18$ girls, of whom $12$ have passed all subjects. In total, $10$ students in the class have failed at least one subject.
a) Construct a two-way table (girls/boys versus passed everything/failed at least one subject).
b) If a person is chosen at random, what is the probability that they are a boy and have passed everything?
c) Given that the chosen person has failed at least one subject, what is the probability that she is a girl?
You can do it on paper and send a photo with your answer.
In an urn there are $4$ red balls and $6$ green balls. How many blue balls must be added so that the probability of drawing a red ball is $\frac{1}{4}$? Set up the…
In an urn there are $4$ red balls and $6$ green balls. How many blue balls must be added so that the probability of drawing a red ball is $\frac{1}{4}$? Set up the equation and solve it.
Back to the beginning: in the bag there are $5+3=8$ possible outcomes, so $P(\text{roja}) = \dfrac{5}{8} = 0.625 = 62.5\,\%$ and $P(\text{azul}) = \dfrac{3}{8} = 0.375 = 37.5\,\%$. Since $0.625 > 0.375$, it is more likely to draw a red marble.