7.05 Future predictions and simulations

Predicted outcome

We can use theoretical or experimental probabilities to help us predict the proportion and number of a given outcome we expect in further experiments. For example, if a coin is tossed $100$100 times how many heads would you expect? The actual outcome of $100$100 trials may vary, however, we can make a reasonable prediction that half the tosses should be heads, so we expect $50$50 heads. 

 

Worked example

Example 1

A spinner is divided equally into $8$8 sections, as shown below.

(a) What is the probability of landing on a green star?

$P\left(\text{green star}\right)$P(green star)	$=$=	$\frac{\text{total favourable outcomes }}{\text{total possible outcomes }}$total favourable outcomes total possible outcomes ​
	$=$=	$\frac{3}{8}$38​

(b) If the spinner is spun $145$145 times, how many times would you expect it to land on a green star?

Think: We take the probability of the event $P\left(\text{green star}\right)$P(green star) and multiply it by the number of trials.

Do:

Expected number of green stars	$=$=	$P\left(\text{green star}\right)\times145$P(green star)×145
	$=$=	$\frac{3}{8}\times145$38​×145
	$=$=	$54.375$54.375

 

At this point we need to round appropriately, so we could say that if the spinner is spun $145$145 times we could expect it to be green approximately $54$54 times.

 

 

Practice questions

Question 1

Question 2

 

Simulations

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 when we calculate probabilities concerning card games or experiments with dice.

However, in many 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 based on our experiences and prior knowledge of similar situations. Although these estimated probabilities may lack precision, we still expect them to behave similarly to other probabilities.

 

Worked example

Example 2

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. If we wish to predict the outcome, what other information might we need to take into account?

Think: At first glance, we might estimate that each team has probability $0.5$0.5 of winning the round. However, what other factors may impact the result?

Do: Before making a prediction we may wish to consider:

• The relative strengths of the opponents that each team has defeated in previous rounds
• The fitness and freedom from injury of the players in the two teams
• Is there a home ground advantage at play
• Does the weather or field conditions better suit one team's style of play
• How have they performed in previous matches against each other

 

 

Practice questions

Question 3

Question 4

Question 5