168r. Reviewing Experimental Probability and Simulation

Learning Intentions

• Understand that the theoretical probability of an event can be estimated by running an experiment, and that running more trials generally gives a better estimate
• Calculate the experimental probability of an event given the results of the experiment
• Calculate the expected number of occurrences given a probability and a number of trials
• Design a simulation Use random devices and Interpret the results of running it

Pre-requisite Summary

• Probability is a number between and that describes the likelihood of an event
• Theoretical probability is found from equally likely outcomes in a sample space
• An experiment can be repeated many times in trials
• Experimental probability is based on what actually happens in the trials
• A fraction can compare the number of successful outcomes with the total number of trials
• An expected number is found by multiplying probability by the number of trials
• A simulation uses a random device, such as a coin, die, spinner or random number generator, to model a real situation
• Results from a simulation need to be interpreted in the context of the situation being modelled

Worked Examples

Worked Example 1

A coin is tossed times and lands heads times.

Solve the experimental probability of heads.

Worked Example 2

A bag contains red and blue counters. A student predicts that the theoretical probability of red is .

They run trials and get red times, then run trials and get red times.

Compare the experimental probabilities and comment on which gives a better estimate of the theoretical probability.

Worked Example 3

The probability of rolling a on a fair die is .

Find the expected number of sixes in rolls.

Worked Example 4

The probability of choosing a green counter is .

Find the expected number of green counters in trials.

Worked Example 5

Design a simulation to model whether a family with two children has exactly one girl.

State the random device, the outcomes, and what each outcome represents.

Worked Example 6

A simulation for a game is run times and a win occurs times.

Find the experimental probability of winning and interpret what this suggests about the game.

Problems

Problem 1

A coin is tossed times and lands heads times.

Find the experimental probability of heads.

Problem 2

A student predicts that the theoretical probability of blue is .

They run trials and get blue times, then run trials and get blue times.

Compare the experimental probabilities and comment on which gives a better estimate of the theoretical probability.

Problem 3

The probability of rolling a on a fair die is .

Find the expected number of fives in rolls.

Problem 4

The probability of choosing a yellow counter is .

Find the expected number of yellow counters in trials.

Problem 5

Design a simulation to model whether a family with two children has two boys.

State the random device, the outcomes, and what each outcome represents.

Problem 6

A simulation for a game is run times and a win occurs times.

Find the experimental probability of winning and interpret what this suggests about the game.

Exercises

Understanding and Fluency

Exercise 1.

Find the experimental probability of each event.

a) A coin lands heads times in tosses

b) A die lands on a total of times in rolls

c) A spinner lands on red times in spins

Exercise 2.

Find the experimental probability of each event.

a) A bag is sampled times and a blue counter is chosen times

b) A basketball player scores a goal times in shots

c) A weather event occurs on days out of

Exercise 3.

Compare each experimental probability with the theoretical probability.

a) Heads in coin tosses gives heads

b) Heads in coin tosses gives heads

c) The theoretical probability of heads is

Exercise 4.

Compare each experimental probability with the theoretical probability.

a) Rolling a in die rolls gives six

b) Rolling a in die rolls gives sixes

c) The theoretical probability of a is

Exercise 5.

Find the expected number of occurrences.

a) Probability in trials

b) Probability in trials

c) Probability in trials

Exercise 6.

Find the expected number of occurrences.

a) Probability in trials

b) Probability in trials

c) Probability in trials

Exercise 7.

A coin is tossed many times.

a) Find the theoretical probability of heads

b) If tosses are made, find the expected number of heads

c) If the coin lands heads times, find the experimental probability

Exercise 8.

A fair die is rolled many times.

a) Find the theoretical probability of rolling an even number

b) If rolls are made, find the expected number of even numbers

c) If an even number occurs times, find the experimental probability

Exercise 9.

Describe a simulation for each situation.

a) Modelling whether a day is sunny or rainy

b) Modelling whether a person is chosen from Group A or Group B

c) Modelling whether a spinner lands on a winning colour

Exercise 10.

A simulation is run and the following results are recorded.

a) In trials, an event happens times. Find the experimental probability

b) If the theoretical probability is , find the expected number in trials

c) Compare the experimental and theoretical results

Reasoning

Exercise 11.

Explain why experimental probability may be different from theoretical probability in a small number of trials.

Exercise 12.

Explain why increasing the number of trials usually gives a better estimate of the theoretical probability.

Exercise 13.

A student says that if the theoretical probability of heads is , then exactly half of tosses must be heads. Explain why this is incorrect.

Exercise 14.

Explain why the expected number of occurrences does not guarantee that this exact number will occur in an experiment.

Exercise 15.

A student uses a simulation with outcomes that are not equally likely to model a fair event. Explain why this may give misleading results.

Problem-solving

Exercise 16.

A game is won by rolling a on a fair die.

a) Find the theoretical probability of winning

b) Find the expected number of wins in games

c) If a player wins games, find the experimental probability

d) Comment on how close the result is to the theoretical probability

Exercise 17.

A weather model predicts rain with probability on a certain kind of day.

a) Find the expected number of rainy days in such days

b) A simulation gives rain on of the days

c) Find the experimental probability

d) Interpret the result

Exercise 18.

A bag contains counters, and the probability of selecting a red counter is .

a) Find the expected number of red counters selected in trials

b) A student actually gets red times

c) Find the experimental probability

d) Decide whether the result is lower or higher than expected

Exercise 19.

Design a simulation to model whether a family with three children has at least two girls.

a) Choose a random device

b) Describe how each outcome represents the situation

c) State what would count as a success

d) Explain how running many trials would help estimate the probability

Exercise 20.

A school designs a simulation for a carnival game.

A player wins if the spinner lands on gold.

The spinner has equal sections and are gold.

a) State the theoretical probability of winning

b) Find the expected number of wins in plays

c) If the simulation gives wins, find the experimental probability

d) Interpret whether the simulation result seems reasonable

Potential Misunderstandings

• Thinking experimental probability must always equal theoretical probability
• Believing a small number of trials gives a perfectly reliable estimate
• Confusing experimental probability with theoretical probability
• Calculating experimental probability using the wrong denominator instead of the total number of trials
• Thinking the expected number is guaranteed to happen exactly
• Forgetting that expected number is found by multiplying probability by the number of trials
• Designing a simulation where the random device does not match the probabilities in the real situation
• Interpreting simulation results without comparing them to the theoretical model
• Assuming that one unusual result means the theoretical probability is wrong
• Forgetting that more trials usually reduce random variation in the estimate