Exercises: Laplace's rule

Calculate the probability of each event, stating the favourable outcomes and total possible outcomes:

1. a) Getting heads when flipping a coin.
2. b) Rolling a $3$ when rolling a die.
3. c) Getting tails when flipping a coin.

We roll a $6$-sided die. We are going to calculate the probability of "rolling a number greater than $4$" step by step:

1. a) How many possible outcomes does the experiment have? Write them down.
2. b) Which outcomes are favourable to the event "greater than $4$"?
3. c) Apply Laplace's rule and write the probability as a fraction.

In a bag there are $4$ red balls and $6$ blue balls, all identical. We draw a ball at random.

1. a) How many balls are there in total?
2. b) What is the probability of picking a red ball?
3. c) And of picking a blue ball?
4. d) Check that the two probabilities above add up to $1$.

In a box there are $10$ trading cards: $4$ are of players and $6$ of team crests. If we pick a card at random:

1. a) What is the probability that it is a player card?
2. b) What is the probability that it is a team crest card?
3. c) Which type of card is more likely to be picked? Why?

In a 1st year ESO class, a raffle is organized with an urn containing $5$ green slips of paper, $7$ blue slips of paper, and $8$ white slips of paper, all of the same size and texture.

Hugo wants to calculate the probability of drawing a green slip at random and writes on the blackboard:

$$P(\text{green}) = \frac{5}{15} = \frac{1}{3}$$

When the teacher asks him, Hugo justifies his calculation like this: "There are $5$ favorable outcomes, which are the green slips, and $15$ possible outcomes, which are the $7$ blue and $8$ white ones that are not green."

1. a) Explain what error Hugo makes when identifying the "possible outcomes" and why his reasoning does not follow Laplace's rule.
2. b) Correct his solution: determine the correct number of possible outcomes and calculate the actual probability of drawing a green slip. Express the result as an irreducible fraction and as a decimal number.
3. c) If we ask for the probability of randomly drawing a red slip from that same urn, what is its numerical value? What is this type of event called?

To celebrate Book Day, $40$ identical bookmarks numbered from $1$ to $40$ have been placed inside a bag in the school library. Whoever draws at random a bookmark with a number that is a multiple of $6$ wins a young adult book.

1. a) Before performing any exact calculation, make an estimate: do you think the probability of winning the book will be greater than, less than, or equal to $0,2$ (that is, $20\%$)? Justify your intuition by thinking about every how many whole numbers a multiple of $6$ appears.
2. b) Write the complete list of all favorable outcomes (the multiples of $6$ between $1$ and $40$) and calculate the exact probability of winning the book using Laplace's rule. Express the result as an irreducible fraction and as a decimal number rounded to the nearest hundredth.
3. c) Compare your exact result with the estimate you made in section a). Was your prediction accurate? Explain why the actual probability does not match dividing $1$ by $6$ exactly when working only up to the number $40$.

From the bag shown in the figure, which contains $10$ balls of three colours, a ball is drawn at random.

1. a) Calculate the probability of picking a red, blue and green ball.
2. b) What is the probability that the ball is not red?
3. c) What is the probability of picking a yellow ball? What is this type of event called?

From the urn in the figure, which contains $10$ counters numbered from $1$ to $10$, a counter is drawn at random.

Calculate the probability of these events:

1. a) Drawing an even number.
2. b) Drawing a multiple of $3$.
3. c) Drawing a number greater than $7$.
4. d) Drawing a single-digit number.

The spinner in the figure has $8$ equal sectors numbered from $1$ to $8$. It is spun once.

Calculate the probability of these events:

1. a) Getting the number $5$.
2. b) Getting an even number.
3. c) Getting a number greater than $5$.
4. d) Not getting $8$.

From a Spanish pack of $40$ cards ($10$ of each suit: coins, cups, swords and clubs), a card is drawn at random. Remember that in each suit there are $7$ cards numbered from $1$ to $7$ and $3$ picture cards (jack, knight and king).

Calculate the probability of these events, giving the irreducible fraction:

1. a) Drawing a coin.
2. b) Drawing an ace.
3. c) Drawing a picture card.
4. d) Drawing a card that is not a cup.

For a recreational mathematics workshop, you have an empty box and counters that feel identical to the touch in three colors: green, red, and black. In total, you must place exactly $24$ counters in the box. Your goal is to design the contents of the box so that these two conditions are met simultaneously:

1. The probability of drawing a green counter at random must be exactly $\frac{1}{3}$.

2. The probability that the counter drawn at random is not black must be $\frac{3}{4}$.

1. a) Deduce how many green, red, and black counters you must put into the box to meet both conditions. Explain step by step how you obtain each quantity.
2. b) With the box already configured, calculate the probability of drawing a red counter. Express the result as an irreducible fraction.
3. c) Without removing any of the initial $24$ counters, how many black counters would you have to add to the box so that the probability of drawing a black counter becomes exactly $\frac{1}{2}$? Justify your answer by setting up the relationship between the new favorable outcomes and the new possible outcomes.

According to the historical records of the State Meteorological Agency (AEMET) for a town in northern Spain during the month of November (which has $30$ days in total), the days were distributed as follows:

• There were $12$ days with rain.

• There were $10$ overcast days, but with no recorded precipitation.

• The remaining days of the month were completely clear.

An active tourism company must randomly choose a date from that month to carry out a pilot hiking route.

1. a) Calculate the probability that the randomly chosen day was a clear day. Express the result as an irreducible fraction and as a percentage rounded to the nearest tenth.
2. b) What is the probability that it did not rain on the chosen date? Show how to obtain this value through two different ways: by adding the direct favorable outcomes and by calculating it using the complementary event.
3. c) The company coordinator states: "By choosing a day in November at random, we have more than a $65\%$ chance that it will not rain." Determine with reasoning whether the coordinator is right or wrong based on your calculations.

A bag contains $12$ balls in total, red and blue. We know that the probability of picking a red ball at random is $\frac{1}{3}$.

1. a) How many red balls are in the bag? And how many blue ones?
2. b) How many blue balls would need to be added to the bag so that the probability of picking a red ball becomes $\frac{1}{4}$?
3. c) Without adding or removing balls from the initial state, which is more likely: picking a red ball or not picking one? Calculate both probabilities.

At the fair there is a spinner with $4$ colours, but its sectors are not equal, as shown in the figure.

A student claims: "Since there are $4$ colours, the probability of each one is $\frac{1}{4}$."

1. a) Explain why their reasoning is incorrect. What condition must be met to apply Laplace's rule?
2. b) Estimate, by looking at the sectors, the probability of each colour, expressing it as a fraction.
3. c) Check that the four probabilities you have given add up to $1$.

We have two bags. Bag A contains $3$ red balls and $5$ blue ones; bag B contains $5$ red ones and $7$ blue ones. We want to pick a red ball.

1. a) Calculate the probability of picking a red ball from each bag.
2. b) In which bag is it more likely to pick a red ball? Is it enough to look at which bag has more red balls? Reason your answer.
3. c) How many red balls would need to be added to bag A (without removing any) so that the probability of picking a red ball is $\frac{1}{2}$?

A $12$-sided die (dodecahedron) has its faces numbered from $1$ to $12$. We roll it once.

1. a) Calculate the probability of rolling a multiple of $4$.
2. b) Calculate the probability of rolling a number greater than $8$.
3. c) Express the two probabilities above as an irreducible fraction, as a decimal and as a percentage.
4. d) What is the probability of not rolling a multiple of $4$? What is the sum of the probabilities of "rolling a multiple of $4$" and "not rolling it"?

Two fair $1$-euro coins are tossed in the air. We want to calculate the probability of event $A$: "getting at least one head." Three 1st year ESO students propose different methods to solve it:

• Alba's method: "The possible outcomes according to the number of heads that can come up are three: $0$ heads, $1$ head, or $2$ heads. Since we are looking for at least one head, the favorable outcomes are two ($1$ head or $2$ heads). By Laplace's rule, $P(A) = \frac{2}{3}$."

• Bruno's method: "I write the sample space detailing each coin: $\{CC, CX, XC, XX\}$. There are $4$ possible outcomes and they all have the same probability of occurring. The favorable outcomes with at least one head are $\{CC, CX, XC\}$, which are $3$. By Laplace's rule, $P(A) = \frac{3}{4}$."

• Clara's method: "I calculate the probability of the complementary event: the opposite of 'getting at least one head' is 'getting no heads' (that is, getting two tails, $XX$). Since there is only $1$ outcome out of the $4$ possible ones, $P(\text{no heads}) = \frac{1}{4}$. Then I subtract from certainty: $P(A) = 1 - \frac{1}{4} = \frac{3}{4}$."

1. a) Analyze Alba's proposal: explain precisely why her application of Laplace's rule is flawed and what fundamental condition regarding the possible outcomes she has ignored.
2. b) Compare Bruno's and Clara's methods. If instead of tossing $2$ coins, $6$ coins were tossed simultaneously, which of the two methods would be much more efficient to calculate the probability of getting at least one head? Justify your answer by explaining the difficulty that the other method would present.
3. c) Use the more convenient method of the two above to calculate the probability of getting at least one head when tossing $3$ fair coins.

Based on a sample from the National Statistics Institute (INE) on sustainability and recycling, $1200$ households in a Spanish city are analyzed to study source separation of waste. The data obtained show the following:

• $720$ households regularly separate plastic packaging (yellow bin).

• $600$ households separate paper and cardboard (blue bin).

• $240$ households do not separate either of these two types of waste.

The city council is going to select a household at random from this sample to present it with an environmental award.

1. a) Organize the information in a two-way contingency table (with the options 'separates plastic / does not separate plastic' versus 'separates paper / does not separate paper') and deduce how many households separate both materials simultaneously.
2. b) If a household is chosen at random, calculate the probability of the following events, expressing each result as an irreducible fraction and as a percentage:

1. That the household separates both plastic and paper.

2. That the household separates at least one of the two types of waste.

3. That the household separates only paper and cardboard, but not plastic.

1. c) A municipal technician claims: "The probability of randomly selecting a household that recycles only plastic is exactly twice that of selecting a household that recycles neither of the two materials." Check numerically whether the technician's statement is true or false and justify your conclusion.