Probability with equally likely outcomes

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 1st Year 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.

Cian 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, Cian 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 Cian 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?

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 $1$.

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.

From a 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.

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 $1$.

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 students propose different methods to solve it:

• Aoife'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, $P(A) = \frac{2}{3}$."

• Cian'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, $P(A) = \frac{3}{4}$."

• Niamh'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 Aoife'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 Cian's and Niamh'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.