Conditional probability experiments and simulations

What is the difference between theoretical probability and relative frequency from a simulation?

A simulation of 200 trials gives 86 successes. Calculate the relative frequency of success.

A bag has 3 red counters and 7 blue counters. Explain how you could simulate one draw using random digits 0–9.

In a with-replacement experiment, why does the probability stay the same after each draw?

In a without-replacement experiment, why must the simulation update the sample space after each draw?

A digital simulation gives estimates 0.46, 0.51 and 0.49 from three equal-sized runs. What single estimate would be reasonable to report?

State one reason a simulation with 50 trials may be less reliable than a simulation with 5000 trials.

Why does a correct simulation of the three-door problem need to follow the host’s exact rule?

Describe one check you would use to decide whether a digital simulation has been set up fairly.

Name one real situation where simulation is useful because an exact probability calculation is difficult.

A spinner has sectors with probabilities 1/2, 1/3 and 1/6. Design a random-number simulation using the integers 1–6, then explain how you would estimate the probability of landing on the first sector at least twice in three spins.

A bag contains 4 red and 6 blue counters. Design two simulations for drawing two counters: one with replacement and one without replacement. Explain exactly how the probability changes between the two models.

A game is won if a player rolls at least one six in four dice rolls. Describe a spreadsheet or coding simulation for 1000 games, including how success is recorded and how the final probability is estimated.

A student simulates a biased coin by assigning heads to digits 0–4 and tails to digits 5–9, but the real coin has P(heads)=0.7. Explain the modelling error and redesign the digit assignment.

Two simulation runs of a chance event give 28 successes out of 100 and 322 successes out of 1000. Compare the estimates and explain which result you would trust more, including limitations.

Design a simulation for the birthday problem in a class of 25 students. State the assumptions, the repeated-trial process and the statistic you would record.

A website simulation claims a game has a 60% chance of winning based on 12 wins out of 20 trials. Analyse whether this is strong evidence and explain what further simulation would improve confidence.

An infection simulation assumes each contact has a 12% chance of transmitting infection. Explain how the simulation could model many people, and state two assumptions that affect the reliability of the result.

A queue at a canteen depends on random arrival times and random service times. Explain how simulation could estimate waiting time, and why one simulated day is not enough.

Create a simulation-based investigation of a chance event of your choice. Include the real event, random device or digital tool, number of trials, success definition, data table headings and how the final estimate would be interpreted.