AGRI 2400 Lecture 10 - Probability

Probability

• the likely outcome of an event we are unsure of, examples all around us since we work with the natural world, and there is always uncertainty due to variation

Basic Probability Rules

• the probability of any event is bound between 0 and 1

  • if an event is certain to occur, its probability is 1

  • if an event is certain not to occur, its probability is 0

• the probability of any event is the number of outcomes resulting in that event divided by the total number of possible outcomes

  • e.g. flipping a coin

Probability Addition Rule

• when outcomes are mutually exclusive, the probability of obtaining any particular result is the sum of their seperate probabilities

• e.g. rolling a 6 sided die

  • probability of rolling any single number is 1/6

  • probability of rolling any number is 6/6

Probability Multiplication Rule for Independent Events

• when the outcome of one event is independent of the outcome of another event, multuply those probabilities to determine the joint probability of both events occurring

Probability Conditional Rule

• when the outcome of one event is not indep from another

• e.g. what is the probability of correctly predicting the numbers of the first 2 consecutively drawn lottery balls?

• can be formally written as: P(a,b) = P(a) X P(b|a)

Probability Helps

• make decisions based on your results

• statistical tests determine: the probability of obtaining the observed difference, or an even more extreme difference, among the samples we measured if the null hypothesis is true

• once that probabiliy is known, you can use it to make a decision about that difference (real or not) using criteria that are uniformly used and understood

Probability and Statistical Tests

• e.g. a bag with 5000 black beads and 5000 white beads (randomized) P(black) =0.5, P(white)=0.5

Probability Helps Make Result Based Decisions

• if the probability of the observed outcome and any more extreme departures possible from the expected outcome, based on null hypothesis, is less than 5% (p<0.05), then it is appropriate to conclude that the observed difference is statistically significant (i.e. that outcome is quite unlikely to occur simply due to random chance)

Null Hypothesis

• there are equal numbers of black and white beads in the bag

Alternate Hypothesis

• there are unequal numbers of black and white beads in the bag

Hypotheses Bases on Results

• the difference between the outcome (6 B) and the expected result (even mix of black and white) has such a low probability of occurring, if the null hypothesis is true, it would be considered statistically significant

• would therefore reject the null hypothesis and conclude that the sample did not come from a pop containing equal numbers of black and white beads P

P Values

• probabilities on which statistical decisions are made are called p-values

  • the p-value measures the probability of obtaining the observed difference, or an even more extreme difference, among the samples we measured if the null hypothesis is true

  • most studies use a 5% cut-off value for decision making

    • i.e. if the p-value is less than 0.05, then there is sufficient evidence to reject the null hypothesis; id the p-value is greater than 0.05 null hypothesis is retained

    • cut off value is called the alpha level

Type 2 Error

• inappropriately fail to reject a null hypothesis that is false

• e.g. telling a pregnant women shes not pregnant

Type 1 Error

• if you inappropriately reject a null hypothesis that is true

• e.g. telling an old man hes pregnant

Reporting P-Values

• report precise values when available, unless very small (<0.00001)

  • when precise values are not available report with reference to your selected alpha

• p values should only be provided after the results/conclusions are written