Laplace's rule
In a bag with 6 red balls and 2 blue ones, which is easier to pick, a red one or a blue one? You can tell at a glance, but maths lets us say it with an exact number.
Favourable outcomes and possible outcomes
When all the outcomes of an experiment are equally likely (a fair die, a balanced coin, balls that are identical except for their colour), we can use Laplace's rule:
$$P(A) = \frac{\text{number of outcomes favourable to } A}{\text{number of possible outcomes}}$$Try different numbers of balls and see how the frequencies get closer to the calculated probability:
Calculate the probability of each event, stating the favourable outcomes and total possible outcomes: a) Getting heads when flipping a coin. b) Rolling a $3$ when…
Calculate the probability of each event, stating the favourable outcomes and total possible outcomes:
- a) Getting heads when flipping a coin.
- b) Rolling a $3$ when rolling a die.
- 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: a) How many possible outcomes does the experiment…
We roll a $6$-sided die. We are going to calculate the probability of "rolling a number greater than $4$" step by step:
- a) How many possible outcomes does the experiment have? Write them down.
- b) Which outcomes are favourable to the event "greater than $4$"?
- c) Apply Laplace's rule and write the probability as a fraction.
Probability with dice and coins
In a bag there are $4$ red balls and $6$ blue balls, all identical. We draw a ball at random. a) How many balls are there in total? b) What is the probability of picking…
In a bag there are $4$ red balls and $6$ blue balls, all identical. We draw a ball at random.
- a) How many balls are there in total?
- b) What is the probability of picking a red ball?
- c) And of picking a blue ball?
- 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. a) Calculate the probability of picking a red, blue and green…
From the bag shown in the figure, which contains $10$ balls of three colours, a ball is drawn at random.
- a) Calculate the probability of picking a red, blue and green ball.
- b) What is the probability that the ball is not red?
- 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: a) Drawing…
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:
- a) Drawing an even number.
- b) Drawing a multiple of $3$.
- c) Drawing a number greater than $7$.
- d) Drawing a single-digit number.
Back to the start: In total there are $6 + 2 = 8$ possible balls; the probability of picking a red one is $\dfrac{6}{8} = \dfrac{3}{4} = 0{,}75$ and the probability of picking a blue one is $\dfrac{2}{8} = \dfrac{1}{4} = 0{,}25$, so picking a red ball is exactly three times as likely as picking a blue one.