An Introduction to Probability

The Theory of Chance!

Consider these statements-the Vikings will probably win the basketball tonite.-it is not likely that it will rain today.-I will probably pass the class-It’ s almost certain I will do well in this chapter.

Each of these statements indicates a likelihood or chance of a particular event happening.

We use percentages to indicate the chance. i.e.0% we believe event will not occor

100% we believe enent is certain to occur.

ProbabilityUse decimals or fractions to represent chance in Mathematics. Zero% 0 100% 1 33.33% 0.333 1/3

Coin toss. Probability it falls on heads is 50% or ½ or 0.50.

In mathematics we writeP(heads)= ½ or P(H) = 0.5

ProbabilityProbability value is a measure of the chance of an event happening.

Assingning probabilities is based on either;•Observing past data (experimentatal probability)•Using arguments of symmetry (theoretical probability)

ProbabilityIf A is an eventh with probabilty P(A) then

Less than zero would imply ”less than impossible.”More than 1 would imply ”more than certain.”•P(A) = 0 event cannot occur•P(A) = 1 event is certain to occur•P(A) close to 1...............highly likely to occur•P(A) close to 0..............hightly unlikely to occur

0 ( ) 1P A

Probability0 0.5 1

Impossible very unlikely unlikely equally likely very likely extremely likely

likely certain

Probability terminologyNumber of trials- the number of times experiment is repeated.Outcomes- different results possible for 1 trial (heads or tails)Frequency- number of times an outcome is observedRelative frequency- frequency of that outcome divided by total number of trials

Relative frequency = frequency number of trials

The relative frequnency of an even is an ESTIMATE of its probability.

exampleTossing a tin can 250 times, it comes to rest onend 37 times. We say,•The nubmer of trials is 250•The outcomes are ends and sides. •The frequency of ends is 27 and sides is 213.•The relative frequency of ends is 37/250 (about 0.148)•The relative frequency of sides is 213/250 (about 0.852)

•P(end) ~ 0.148•P(side) ~ 0.852

Real worldAn insurance company receives9573 claims from 213,829 clients.

The probability of a client making a claim in the next year can be predicted by the relative frequency;

9537/213829 ~ 0.0446 ~ 4.46%.

This well help company calculate its premiums for the following year.