Mixed Statistics so far Find estimates for the mean and standard deviation of the amount spent. The students’ teachers want to know the mean and Standard deviation in Euros (1 GBP = EUR) If the trip cost 120 Euros plus spending what are the mean and standard deviation of the total cost for students?

Aims: To learn the basics of probability and associated notations.

Outcomes Name: Know what “probability” problems are. Meet the idea of “unions” and “intersctions”. Also “equally likely outcomes”. “Multiplication law”. Describe: How probability is measured and how basic probabilities can be calculated for equally likely outcomes. Explain: How tree diagrams work and can be used to find the probability of successive events. Similarity between this and the multiplication law.

Maths S1 Textbook Ex 2A and 2B Next Lesson: Probabilities of compound events and conditional probabilities. Christiaan Huygens : Dutch Physicist and Astronomer (as well as Mathematician) he discovered the rings of Saturn and was the first person to publish a book on probability.

Basic Probability Ideas – Probability Scale Probability is measured on a scale from 0 to 1 as a fraction/decimal or 0 to 100% as a percentage.

Some Notation Probability has some notations that you may not have seen before. We often describe events (things that can happen) with letters. A: A person selected at random has brown hair. E: A student selected is studying Chemistry. P: A dice roll is a prime number.

Some Notation A: A person selected at random has brown hair. This can then be combined with other notations… n(A) represents the number of the possible outcomes where A is true. P(A) is the probability that A happens. A’ is called the “complement” of A and is the opposite of A (a person selected at random does not have brown hair. It can be used with the above too P(A’) is the probability A does not happen.

Getting Used to It We are going to conduct an experiment. Throughout this we will write experimental probabilities.

Equally Likely Probability is easy to understand and assign where we are dealing with equally likely outcomes. E.g. If a 12 sided die is rolled what is… P(12) P(< 5) P(Prime) P(5’) n(even)

Some Words Mutually Exclusive: Exhaustive:

Our Class... In a class of __ students. There are __ with brown hair. A student is selected at random… Let A be the event a student has brown hair. On a whiteboard what are n(A) P(A) n(A’) P(A’) P(A’)=1-P(A)

Basic Probability

More on Complements Since we measure probability on a scale from 0 to 1 and something either occurs or does not then… P(A) + P(A’) = 1 P(A) = 1- P(A’) P(A’) = 1 – P(A)

Successive Events At GCSE you will have encountered successive events (probabilities based on two different events) in the form of tree diagrams. When dealing with successive events we have some additional notation to see. A  B called the union is the event of A or B (or both) happening (at least one of A or B) A  B called the intersection is the event where A and B both happen. These can be combined with notation seen earlier P(A  B ), n(A  B), A’  B, (A  B)’

Tree Diagrams A six sided dice is rolled and coin is flipped.

Tree Diagrams

Multiplication Law The idea of multiplying along the branches can be applied to problems without a diagram. For successive independent events P(A  B) = P(A) x P(B) E.g. A dice is rolled three times what is the probability that all the rolls are even numbers?

Multiplication Law Careful

Addition Law The last example saw us having to sum probabilities as more than one possible outcome (from the invisible tree diagram) represented a success. This is an application of the addition law. For mutually exclusive events A and B P(A  B) = P(A) + P(B)

Some Addition Law (some not)

The “At Least One” Trick A fair 8 sided die is thrown 6 times what is the probability at least on 8 is rolled?

At Least One