Probability

Lesson

When we have enough information about the ways in which outcomes can occur in an experiment, we can assign probabilities to them by doing calculations involving counting or measuring. This is the case, for example, when we calculate probabilities concerning card games or experiments with dice.

However, in most practical situations it is not possible to gather enough information about the causes and effects involved in order to be able to calculate the probabilities of the possible outcomes with any degree of certainty. Even when the mechanisms governing some process are well understood, the mathematics needed to calculate the probabilities may be difficult or impossible.

For these reasons, the likelihoods of various outcomes concerning such things as the weather, the climate, the results of sporting events or the occurrence of a disease are investigated by statistical sampling, by observational studies or by simulations, possibly involving computer modelling. It is assumed that the relative frequencies of occurrence of the various outcomes observed in a study correspond reasonably well to the supposed underlying mathematical probability.

In practice, for many everyday situations, we estimate probabilities subjectively based on our experiences and prior knowledge of similar situations. Although such estimated probabilities may lack precision, we nevertheless expect them to behave similarly to other probabilities, in accordance with the usual mathematical rules.

Suppose two teams in a sporting competition have each won three out of four of their previous games and they are about to play against each other. We wish to predict the outcome.

At first glance, we might estimate that each team has probability $0.5$0.5 of winning the round. However, after considering the relative strengths of the opponents that each team has defeated in previous rounds, we might modify the probability estimate in favour of one team or the other. We might also consider the fitness and freedom from injury of the players in the two teams before making a prediction.

When studying the results of a baseball team, previous results of performance are looked at. Other factors that could complicate the calculation of the probability of the team winning their next game could be:

Select all the correct options.

Players' conditions

AHow well they played last week

BHow well another team they haven't played against has been playing in the competition

CWeather

DPlayers' conditions

AHow well they played last week

BHow well another team they haven't played against has been playing in the competition

CWeather

D

The results of a particular study state that if you exercise for at least $2$2 hours a week, the risk of developing heart disease is significantly reduced. What factors might elevate the risk of heart disease for someone who exercises more than $2$2 hours a week?

Select all the correct options.

Having a partner who has a heart disease

ATheir uncle having had heart disease

BA poor diet

CHaving previously had a heart infection

DHaving a partner who has a heart disease

ATheir uncle having had heart disease

BA poor diet

CHaving previously had a heart infection

D

In a new study scientists have determined that it is very likely that a recent widespread drought has increased global warming.

Sceptics looking to argue against this claim could point to which other possible contributing factors? Select all correct answers.

Governments reducing green energy initiatives.

AA steady increase in global temperatures in the lead up to the drought.

BIncreased burning of fossil fuels.

CA rise in global temperatures during previous droughts.

DGovernments reducing green energy initiatives.

AA steady increase in global temperatures in the lead up to the drought.

BIncreased burning of fossil fuels.

CA rise in global temperatures during previous droughts.

D

Investigate situations that involve elements of chance: A calculating probabilities of independent, combined, and conditional events B calculating and interpreting expected values and standard deviations of discrete random variables C applying distributions such as the Poisson, binomial, and normal

Apply probability concepts in solving problems