PROBABILITY Experimental Probability n Experimental approach n n

PROBABILITY

Experimental Probability n Experimental approach n n n 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 n Or: Probability = 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 1 st 10 2 D 3 U 4 U 5 U 6 D 7 U 8 D 9 D 10 D U 2 nd 10 3 rd 10 4 th 10 5 th 10 6 th 10 U= point up D= point down Row total U 5

Probability as long run relative frequency Number of trials 1 2 3 4 5 10 20 30 40 50 60 70 80 90 100 Frequency of Us 0 1 2 3 3 5 Relative frequency of Us Fraction Decimal 0 1/2 2/3 3/4 3/5 5/10 0 0. 5 0. 666 0. 75 0. 6 0. 5

Probability of drawing pin landing point up Long run relative frequency

Theoretical probability n n 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.

Theoretical probability n n 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

Playing Cards 4 suits 13 cards in each suit n Spades Hearts n Diamonds Clubs 52 cards in a deck of cards + 2 jokers

Dice n n n One die, many dice Six sides Six numbers 1, 2, 3, 4, 5, 6

Coin toss n Heads or tails

Definitions n n n 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

Example n n n 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 n n n 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)

Probability functions n n n 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 x 0 1 2 3 P(X =x) n The total of the probabilities is always 1

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

Venn diagrams n n n 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’

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

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

Probability rules P(A E. g. n P(A B) B) = P(A) + P(B) – P(A = 0. 75 + 0. 8 – 0. 65 = 0. 9 B)

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) 0. 1 P(B) P(A∩B) =0. 65 0. 15 The probability that they will eat neither is

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

Mutually exclusive events n 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 n 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 n

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

Contingency tables n A contingency table can show frequencies: Drink Tea Do not drink tea Total Drink Coffee 51 77 128 Do not drink coffee 18 51 72 Total 69 131 200 or probabilities: 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 Total 0. 345 0. 655 1

Probability trees n n 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