Laplace's Rule
At a corner store there is a bag with 5 passion fruit candies and 3 strawberry candies, all wrapped the same way. If Yaritza takes one without looking, which is more likely: passion fruit or strawberry? You can tell just by looking, but with math we can give an exact number.
Favorable cases and possible cases
When all the outcomes of an experiment are equally likely (a fair die, the cards in a deck, balls that are identical except for color), we use Laplace's rule:
$$P(A) = \frac{\text{number of favorable cases for } A}{\text{number of possible cases}}$$Try different numbers of balls and see how the frequencies get closer to 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.
The letters of the word PROBABILIDAD are written on cards, and one card is drawn at random. Find the probability that it is: a) A vowel. b) The letter B. c) A consonant.
The letters of the word PROBABILIDAD are written on cards, and one card is drawn at random. Find the probability that it is:
a) A vowel.
b) The letter B.
c) A consonant.
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?
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.
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?
Laplace with numbers, cards, and data tables
Many exercises are solved by counting carefully. With events like "A or B," count the cases that meet A, B, or both, without repeating the ones that meet both.
Two-way tables with data
If you are given a table with groups, the total of the table is the number of possible cases. If you are told "given that…", the possible cases are reduced to that row or column.
A card is drawn from a Spanish deck of $40$ cards. Find the probability that it is: a) Coins (Oros*). b) An ace. c) A face card (jack, knight, or king). d) The king of…
A card is drawn from a Spanish deck of $40$ cards. Find the probability that it is:
a) Coins (Oros).
b) An ace.
c) A face card (jack, knight, or king).
d) The king of cups.
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 $5$.
The students at a school who do extracurricular sports are distributed as follows: | | soccer | swimming | tennis | |---|---|---|---| | $6.º$ Primary | $14$ | $7$ | $4$…
The students at a school who do extracurricular sports are distributed as follows:
| soccer | swimming | tennis | |
|---|---|---|---|
| $6.º$ Primary | $14$ | $7$ | $4$ |
| $1.º$-$2.º$ Secondary | $16$ | $4$ | $15$ |
A student is chosen at random. Calculate the probability that:
a) They are in Primary.
b) They do swimming.
c) They are in Primary and play tennis.
d) Given that they are in Secondary, 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.
Laplace in compound experiments
With two dice, two coins, or a coin and a die, first count all the possible cases using a two-way table or a tree diagram, and then apply Laplace's rule.
Without replacement: the possible cases change
If you draw two cards without putting the first one back, when you draw the second there is one fewer card, and the number of favorable cases may change.
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 $5$.
Two dice are rolled and the sum of the scores is recorded. Use a sample space table to calculate: a) $P(\text{sum } 7)$ b) $P(\text{sum greater than } 9)$ c)…
Two dice are rolled and the sum of the scores is recorded. Use a sample space table to calculate:
a) $P(\text{sum } 7)$
b) $P(\text{sum greater than } 9)$
c) $P(\text{even sum})$
d) $P(\text{sum other than } 7)$
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.
To enter a game at the school fair, you must roll a die and get a higher score than the 'guardian', who rolls another die before you. a) Construct a sample space table…
To enter a game at the school fair, you must roll a die and get a higher score than the 'guardian', who rolls another die before you.
a) Construct a sample space table with the $36$ possible matchups.
b) What is the probability that you enter the game?
c) What is the probability of tying with the guardian?
d) If the rule were 'score at least twice as many points as the guardian', what would be the probability of entering?
You can do it on paper and send a photo with your answer.
Two cards are drawn from a standard Spanish deck of $40$ cards, without replacing the first card in the deck. a) What is the probability that the first card is coins? b)…
Two cards are drawn from a standard Spanish deck of $40$ cards, without replacing the first card in the deck.
a) What is the probability that the first card is coins?
b) If the first card was coins, what is the probability that the second card is also coins?
c) What is the probability that both cards are coins?
d) And what is the probability that neither card is coins?
Laplace with spinners with unequal sectors
If a spinner does not have equal sectors, we use the angle of each sector, which is proportional to the probability.
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.
Probability and equations
Sometimes the probability is known and you are asked for an unknown quantity. Set up Laplace's formula as an equation.
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.
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 $5+3=8$ candies, so $P(\text{passion fruit}) = \dfrac{5}{8} = 0.625$ and $P(\text{strawberry}) = \dfrac{3}{8} = 0.375$. Since $0.625 > 0.375$, it is more likely to take a passion fruit candy.