Exercises: Laplace's rule

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 green ball.

c) Drawing a yellow ball.

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 picture card (jack, knight, or king).

d) The king of cups.

A spinner is divided into $8$ equal sectors numbered from $1$ to $8$. Find the probability that the arrow stops on:

a) $5$.

b) A multiple of $3$.

c) A number greater than $5$.

d) A number less than $9$.

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 $5$.

In a $2.º$ ESO class, there are $14$ girls and $16$ boys. A person is chosen at random to read aloud.

a) What is the probability that it is a girl?

b) And that it is a boy?

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.

A tile is chosen at random from a complete set of $28$ dominoes. Find the probability that it is:

a) Double $6$.

b) Any double tile (there are $7$).

c) A tile containing the number $0$ (there are $7$).

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 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.

A fair die is rolled. Find the probability of rolling:

a) A multiple of $3$.

b) A prime number.

c) A number less than $7$.

d) A number greater than $6$.

A ball is drawn from a lottery drum containing balls numbered from $1$ to $12$. Find the probability that it is:

a) An even number.

b) A number greater than $8$.

c) A factor of $12$.

d) A multiple of $5$.

In Mathematics class, Lucía presents a spinner divided into three colored sectors: a red one with a central angle of $90^\circ$, a blue one also of $90^\circ$, and a yellow one of $180^\circ$. To calculate the probability that the arrow stops on the red color, Lucía states: "Since there are three possible outcomes (red, blue, and yellow) and red is one of them, by Laplace's rule the probability is $P(\text{red}) = \frac{1}{3}$".

a) Explain in detail the mistake Lucía makes when directly applying Laplace's rule by only counting the colors.

b) Calculate the actual probability that the arrow points to the red sector using the relationship between the sector's angle and the complete circle. Express the result as an irreducible fraction and as a percentage.

c) Use the formula for the complementary event to calculate the probability that the arrow does not stop on the yellow sector, that is, $P(\overline{\text{yellow}})$.

A Mathematics teacher prepares an opaque bag with $20$ tokens that feel identical to the touch, but in three different colors: red, green, and blue. So that her students figure out how many tokens there are of each color without looking inside, she provides them with the following clues:

• When drawing a token at random, the probability that it is red is exactly $0,4$.

• The probability of drawing a blue token is twice that of drawing a green token.

a) How many red tokens does the bag contain?

b) Set up a reasoned deduction to determine the number of green tokens and the number of blue tokens the bag contains.

c) Check numerically that the sum of the elementary probabilities of the three colors satisfies the fundamental property of probability.

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?

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) $P(\text{sum other than } 7)$

Two coins are tossed. Calculate the probability of getting:

a) Two heads.

b) One head and one tail, in any order.

c) At least one head.

d) No heads.

A card is drawn from a Spanish deck of $40$ cards. Calculate the probability that it is:

a) Coins or a picture card.

b) Cups or an ace.

c) Neither a picture card nor an ace.

d) Clubs or a number card less than $4$ (the ace counts as $1$).

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 students at a school who do extracurricular sports are distributed as follows:

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 ESO, they play tennis.

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 $8$.

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 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.

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.

For a Geography exam, students must place the $17$ autonomous communities on an outline map. Ricardo only knows the location of $9$.

a) If he is asked to locate one at random in the exam, what is the probability that it is one of those he knows?

b) If he gets one he does not know and guesses its location among the remaining $8$, what is the probability that he gets it right?

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.

Two fair six-sided cubic dice (numbered from $1$ to $6$) are rolled simultaneously, and the scores obtained are added together. Event $A = \text{«the sum is less than or equal to } 4\text{»}$ is defined.

a) Before doing any mathematical calculation, make an intuitive estimate: do you think the probability of event $A$ occurring will be greater than, less than, or approximately equal to $25\,\%$? Write a sentence justifying your first impression.

b) Draw a $6 \times 6$ two-way table showing the $36$ possible outcomes for the sum of the two faces.

c) Apply Laplace's rule to your table to calculate the exact value of $P(A)$. Express the result as an irreducible fraction and as a percentage with one decimal place.

d) Compare the mathematical result obtained with your estimate from part a) and explain why many people intuitively tend to overestimate the frequency of low sums in this experiment.

At a Cercanías station in Madrid, the arrivals of $80$ trains are recorded during one morning to analyze the punctuality of three lines (C-3, C-4, and C-5). The recorded data are:

• For line C-5, $48$ trains run, of which $42$ arrive on time and $6$ are delayed.

• For line C-3, $20$ trains run, with $15$ on time and $5$ delayed.

• For line C-4, $12$ trains run, with $9$ on time and $3$ delayed.

a) Organize all the data into a two-way contingency table showing the lines in the rows and punctuality (on time / delayed) in the columns, adding the marginal totals.

b) If a commuter boards a train chosen at random from all those recorded that morning, calculate the probability that they travel on a line C-5 train and that it arrived on time. Also calculate the probability that they board a train that suffers a delay (regardless of the line).

c) A passengers' association states: "Line C-5 runs twice as poorly as C-4 because it accumulates $6$ delayed trains and C-4 only $3$". Use Laplace's rule to calculate and compare the probability that a train suffers a delay within line C-5 with the probability that it suffers a delay within line C-4. Reason whether the association's complaint is mathematically justified.

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 dice are rolled and the product of the scores is recorded.

a) Construct the sample space table with the $36$ products.

b) Calculate the probability that the product is even.

c) Calculate the probability that the product is a multiple of $5$.

d) Calculate the probability that the product is greater than $20$.

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 equation and solve it.

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 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?

A tetrahedral die has its faces numbered $1$, $2$, $2$ and $3$, and another tetrahedral die has $2$, $3$, $3$ and $4$. Both are rolled and the product is recorded.

a) Construct the sample space table with the $16$ possible products.

b) Calculate the probability that the product is even.

c) Calculate the probability that the product is $6$.

You can do it on paper and send a photo with your answer.

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.

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?

In a cooperative board game, to activate a protective spell, $3$ fair six-sided cubic dice are rolled at the same time. The condition for the spell to work is to roll at least one $6$ among the three dice. Two students discuss how to calculate the probability of success:

• Sofía suggests: "It is much faster to first calculate the probability of not rolling any $6$ on any of the dice and then apply the complementary event property: $P(\text{win}) = 1 - P(\text{no } 6)$"

• Daniel suggests: "It is better to calculate the favorable outcomes by adding three different situations: rolling exactly one $6$, rolling exactly two $6$s, or rolling three $6$s".

a) Determine the total number of possible outcomes when rolling the three six-sided dice simultaneously.

b) Work through Sofía's method step by step: calculate how many outcomes do not contain any $6$, obtain the probability of not winning, and deduce the final probability of winning as an irreducible fraction and an approximate percentage.

c) Work through Daniel's method: count in a reasoned and organized way how many possible outcomes correspond to rolling exactly one $6$, how many to exactly two $6$s, and how many to three $6$s. Verify that the total sum of favorable outcomes matches Sofía's result exactly.

d) Justify with reasons which of the two methods you would choose if the game rule changed and $5$ dice were rolled instead of $3$, evaluating the difficulty of counting in each approach.

A youth association designs a $360^\circ$ circular spinner to organize a charity raffle and raise funds for a social cause. The spinner has four sectors: Gold, Silver, Bronze, and White (no prize). The guidelines agreed upon for its design are:

• The probability of landing on the "Gold" sector must be exactly half that of landing on the "Silver" sector.

• The "Bronze" sector must cover a central angle of $90^\circ$.

• The probability of not winning a prize ("White" sector) must be exactly $0,4$.

• Each spin will cost participants $1$ euro, and a total of $600$ spins are expected to take place throughout the day.

a) Deduce the central angle in degrees ($^\circ$) and the theoretical probability corresponding to each of the four sectors of the spinner.

b) The established prizes are: the "Gold" sector awards a $5$-euro voucher, the "Silver" sector awards a $2$-euro voucher, the "Bronze" gives a commemorative souvenir that cost the association $0,50$ euros, and the "White" gives nothing. Calculate the total expected expenditure on prizes if the $600$ planned spins are carried out.

c) Compare the expected revenue from selling the $600$ spins with the estimated prize expenditure: will the association manage to raise profits with this design, or will it incur losses? Justify your numerical answer and propose a concrete change in the distribution of the sectors to correct the situation while keeping all four types of prizes.