PROBABILITY

Experimental Probability Experimental approach Probability can be worked out by observation over a large number of trials. This is appropriate when the conditions of the experiment remain stable Probability = number of favourable outcomes total number of trials Or: Probability = Long run relative frequency L1

Relative Frequency Calculating & graphing long run relative frequency Concept: How many trials before the experimental probability is close to the actual probability? Relative Frequency Calculating & graphing long run relative frequency

Probability as long run relative frequency 1 2 3 4 5 6 7 8 9 10 Row total U 1st 10 D U 2nd 10 3rd 10 4th 10 5th 10 6th 10 L1: Students have this table in their handout booklet. Demonstrate how to fill in the table U= point up D= point down

Probability as long run relative frequency Number of trials Frequency of Us Relative frequency of Us Fraction Decimal 1 2 1/2 0.5 3 2/3 0.666 4 3/4 0.75 5 3/5 0.6 10 5/10 20 30 40 50 60 70 80 90 100 L1: Show students this slide so they can fill in their tables. Ask them to plot the numbers on the relative frequency graph in their booklet.

Probability of drawing pin landing point up Long run relative frequency L1: Ask if their graph came out looking like this. Compare the different long run relative frequencies round the class.

Theoretical probability In situations where there are equally likely outcomes, probability can be calculated without having to carry out an experiment. Common examples of equally likely outcomes are from tossing coins, dices or drawing cards out of a pack. L1: Show students the following slides to be sure they know what a deck of cards is. You could give them packs of cards to look at and ask students to find out how many spades , hearts, clubs, diamonds there are. (if you have time!)

Theoretical probability Probability = number of elements in the event number of elements in the sample space Example: A single die is tossed. What is the probability of getting an even number? Solution: P(E) = = = 0.5 L1:

Playing Cards Spades Hearts Diamonds Clubs 4 suits 13 cards in each suit Spades Hearts Diamonds Clubs L1: These slides are in the student booklet. They have been asked these questions Calculate these probabilities for a pack of cards a) P(Red card) = b) P(Diamond) = c) P( an Ace) = 52 cards in a deck of cards + 2 jokers

Dice Six sides Six numbers 1,2,3,4,5,6 One die, many dice L1: Students have been asked to find these probabilities Calculate a) P(six) = Calculate the probability of getting a head P(Head) = b) P( even number) = One die, many dice

Coin toss Heads or tails L1: Calculate the probability of getting a head P(Head) =

Definitions A random experiment is a process with a result which depends on chance A trial is one performance of the experiment An outcome is the result of the experiment The sample space is the set of all possible outcomes of the experiment An event is a subset of the sample space L1:

Example Rolling a single die is a random experiment Rolling the die once is a trial Getting a 4 is one outcome {1,2,3,4,5,6} is the sample space Getting an even number is an event

Random variables A random variable is a variable whose value comes from a random experiment Capital X is used to stand for the name of the variable and small x is used for the value. We write P(X = x) L1:

Probability functions Every random variable has a probability function which associates a probability with each value of x. Example: Three coins are tossed. A random variable X is the number of heads The probability distribution for X is The total of the probabilities is always 1 x 1 2 3 P(X =x) L1: You should ask students to help you work out all the possible combinations. You haven’t taught tree diagrams yet but some students may know how to use them.

Basic Probability A Random experiment - a process which depends on chance -eg rolling a die with six sides A trial - one performance of the experiment - rolling a die once An outcome - the result of the experiment - rolling a 4 The sample space -the set of all possible outcomes - for a die {1, 2, 3, 4, 5, 6} An event -a subset of the sample space - getting an odd number when rolling a die {1,3,5} L2: Revision for the start of lesson 2

Venn diagrams Venn diagrams are a useful for solving probability problems. A circle is used to represent an event A Two or more events can be combined by ‘union’ or ‘intersection’ L2:

Intersection The intersection of two events represents BOTH occurring Read this as ‘A intersection B’ L2:

Union The union of two events represents at least one of the events occurring Union Read this as ‘A union B’ L2:

Probability rules P(A B) = P(A) + P(B) – P(A B) E.g. = 0.9 L2: Students were asked in their booklets “The union of sets A and B can be calculated in two ways. What is the other way?”

Example The probability that an individual will eat apple is P(A)=0.75. The probability that he/she will eat banana is P(B)=0.8. The probability that they will eat both is P(AUB) = 0.65. Calculate that probability that they will eat neither. The probability that they will eat apple or banana or both is P(AUB) = P(A) +P(B)-P(A∩B) = 0.75 + 0.8 -0.65 = 0.9 P(A) P(B) The probability that they will eat neither is L2: P(A∩B) =0.65 0.1 0.15

Complementary events If A is an event, then not A is the complementary event. The is written as A′ A A′ P(A) + P(A′) = 1 L2:

Mutually exclusive events Mutually exclusive events cannot both occur on the same trial. In a Venn diagram they do not overlap. A B P(A B) = 0 So P(A B) = P(A) + P(B)

Independent events When the occurrence of A has no effect on the occurrence of B or vice versa, the two events are independent. The statistical definition of independent events is P(A B) = P(A) P(B) A and B are independent

Parts of the Venn diagram Each part of a Venn diagram represents a different combination of events

Contingency tables A contingency table can show frequencies: or probabilities: Drink Tea Do not drink tea Total Drink Coffee 51 77 128 Do not drink coffee 18 72 69 131 200 Drink Tea Do not drink tea Total Drink Coffee 0.255 0.385 0.64 Do not drink coffee 0.09 0.27 0.36 0.345 0.655 1

Probability trees A probability tree can be used to work out probabilities when several events occur one after the other. Outcomes are written at the end of branches Probabilities are written along the branches Multiply probabilities along the branches