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7.05 Future predictions and simulations

Lesson

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.

 

Remember!

$\text{Expected number of a given outcome}=P\left(\text{outcome}\right)\times\text{Number of trials}$Expected number of a given outcome=P(outcome)×Number of trials

 

Practice questions

Question 1

A car manufacturer found that $1$1 in every $4$4 cars they were producing had faulty brake systems. If they had already sold $5060$5060 cars, how many of those already sold would need to be recalled and repaired?

Question 2

$260$260 standard six-sided dice are rolled.

  1. What is the probability of getting an even number on a single roll of a die?

  2. How many times would you expect an even number to come up on the $260$260 dice?

 

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

 

Remember!

Just because we can develop an expected outcome from a simulation, doesn't mean that the expected outcome is realistic. We always need to consider factors that might complicate the simulation and hence, alter the expected outcome.

 

Practice questions

Question 3

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:

  1. Select all the correct options.

    Players' conditions

    A

    How well they played last week

    B

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

    C

    Weather

    D

Question 4

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?

  1. Select all the correct options.

    Having a partner who has a heart disease

    A

    Their uncle having had heart disease

    B

    A poor diet

    C

    Having previously had a heart infection

    D

Question 5

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.

  1. Governments reducing green energy initiatives.

    A

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

    B

    Increased burning of fossil fuels.

    C

    A rise in global temperatures during previous droughts.

    D

Outcomes

ACMEM152

identify relative frequency as probability

ACMEM153

identify factors that could complicate the simulation of real-world events

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