Laplace's rule
In a bag with 5 red balls and 3 blue ones, which is easier to pull out, a red one or a blue one? You can tell at a glance, but maths lets us put an exact number on it.
Favourable outcomes and possible outcomes
When all the outcomes of an experiment are equally likely (a fair die, cards from a deck, balls identical except for their colour), we can use Laplace's rule:
$$P(A) = \frac{\text{número de casos favorables a } A}{\text{número de casos posibles}}$$Try different numbers of balls and see how the frequencies approach the calculated probability:
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 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 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.
The complementary event
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…
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?
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…
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.
Laplace's rule in compound experiments
With two dice, two coins, or a coin and a die, we first count all the possible outcomes using a two-way table or a tree diagram, and then apply Laplace's rule to those outcomes.
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)…
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) $P(\text{sum other than } 7)$
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 $5$.
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 $5$.
Laplace's rule with spinners with unequal sectors
If a spinner does not have equal sectors, each region is no longer "one outcome": we have to use its angle as a measure of the space it occupies, because the angle is proportional to the probability.
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…
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.
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…
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.
Back to the start: In the bag there are $5+3=8$ possible outcomes, so $P(\text{roja}) = \dfrac{5}{8} = 0{,}625$ and $P(\text{azul}) = \dfrac{3}{8} = 0{,}375$; since $0{,}625 > 0{,}375$, it is more likely that you will pull out a red ball.