Exercises: Laplace's rule

Calculate the probability of each event, stating the favorable 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?

From the bag shown in the figure, which contains $10$ balls of three colors, 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 deck 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 $3$ face 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 face card.
4. d) Drawing a card that is not a cup.

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 $4$ colors, but its sectors are not equal, as shown in the figure.

A student claims: "Since there are $4$ colors, 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 color, 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"?