Compound probability: trees, tables & simulations

A compound event combines two or more separate events, like flipping a coin AND rolling a die. Listing every possible outcome — using a tree diagram, table, or systematic list — lets you assign a probability to each specific combination. "And" events (both things happening) generally multiply probabilities; inclusive "or" events (at least one happening) generally add them, adjusting for any overlap. Relative frequency — the proportion of times an outcome actually occurred in repeated trials or simulations — can be used to estimate a probability when the exact theoretical probability is hard to calculate directly, and gets closer to the true probability as more trials are run.

Key terms

Compound event:

An event made up of two or more separate events combined together.

Relative frequency:

The proportion of times an outcome actually occurred in repeated trials.

Questions

1. 1. A compound event combines:

   • Two or more separate events
   • Only a single event
   • No events at all
   • Only impossible events

2. 2. Flipping a coin AND rolling a die has how many total outcomes?

   • 12
   • 6
   • 2
   • 8

3. 3. An "and" event generally involves:

   • Multiplying probabilities
   • Always adding probabilities
   • Ignoring probability entirely
   • Subtracting probabilities

4. 4. Relative frequency is:

   • The proportion of times an outcome actually occurred in trials
   • Always exactly equal to the theoretical probability
   • A type of graph only
   • Unrelated to probability

5. 5. A tree diagram helps you:

   • List every possible outcome of a compound event
   • Only show a single outcome
   • Avoid calculating probability
   • Replace the need for any data

6. 6. Relative frequency gets closer to the true probability as:

   • More trials are run
   • Fewer trials are run
   • Trials have no effect at all
   • The experiment is stopped immediately

7. 7. The probability of heads AND an even number (from a coin and die) is:

   • 1/4
   • 1/2
   • 1
   • 1/12

8. 8. Rolling two dice, what is the probability of getting a 6 on both?

   • 1/36
   • 1/6
   • 1/12
   • 2/6

9. 9. A bag has a red and blue marble. Drawing one, replacing it, then drawing again — how many total outcomes are there?

   • 4
   • 2
   • 8
   • 1

10. 10. A simulation of 500 coin flips shows heads 240 times. The relative frequency of heads is:

    • 0.48
    • 0.24
    • 0.5
    • 2.4

11. 11. The probability of rolling an even number OR a number greater than 4 on a die (inclusive "or") requires:

    • Adding the probabilities and adjusting for any overlap
    • Only multiplying the probabilities
    • Ignoring one of the two events
    • Only considering odd numbers

12. 12. Why might relative frequency from a large number of trials be a useful estimate when a theoretical probability is hard to calculate directly?

    • Repeated real trials can approximate the true probability even when a clean theoretical calculation is complex or impossible
    • Relative frequency is never a useful method for estimating probability
    • Theoretical probability is always trivially easy to calculate for every situation
    • Simulations and trials provide no useful information about probability

13. 13. Two events without replacement (like drawing cards without putting them back) generally have:

    • Changing probabilities for the second event, since the total pool has decreased
    • Identical probabilities for every draw regardless of what was already drawn
    • No connection between the first and second draw
    • Always exactly the same number of total outcomes as with replacement

14. 14. Why might a tree diagram be more useful than a plain list when there are three or more compound events to consider?

    • It organises outcomes systematically, making it easier to count all combinations without missing any
    • Tree diagrams become useless once there are more than two events
    • A plain list is always clearer than a tree diagram for three or more events
    • Tree diagrams can only ever represent a single event

15. 15. Why might running only 10 trials of a simulated chance experiment give a relative frequency quite different from the true theoretical probability?

    • A small number of trials is more affected by random chance variation than a very large number of trials
    • Small numbers of trials always exactly match the true theoretical probability
    • The true probability changes depending on how many trials are run
    • Trial count has no bearing on how close relative frequency gets to the true probability

16. 16. Why does drawing two cards from a deck WITHOUT replacement require different probability calculations than drawing WITH replacement?

    • Removing a card changes the total number of cards and possibly the number of favourable outcomes remaining for the second draw
    • Replacement never has any effect on probability calculations
    • Both scenarios always produce exactly identical probabilities
    • Only the second draw matters, regardless of whether the first was replaced

17. 17. Why might a casino or insurance company rely on the long-run relative frequency of an event (based on huge amounts of historical data), rather than a single trial, to set their odds or pricing?

    • Relative frequency from an enormous number of past trials provides a very reliable estimate of the true underlying probability
    • A single past trial is always just as reliable as data from millions of historical trials
    • Casinos and insurance companies never actually use probability or historical data
    • Long-run relative frequency has no genuine connection to true probability

18. 18. Why might correctly identifying whether an "or" is inclusive (at least one) or exclusive (exactly one but not both) change how you calculate a compound probability?

    • Inclusive and exclusive "or" require different adjustments for overlapping outcomes, leading to different final probabilities
    • Inclusive and exclusive "or" always produce exactly the same probability regardless of interpretation
    • The type of "or" used has no bearing on how a compound probability should be calculated
    • Only "and" events, never "or" events, require any careful calculation

19. 19. A weather app reports "70% chance of rain" based on historical relative frequency of rain occurring under similar atmospheric conditions. Why is this a legitimate use of probability, even though tomorrow's weather is a one-off event?

    • It estimates the likelihood based on how often rain has historically followed similar conditions, even though any single day's actual outcome is either rain or no rain
    • Relative frequency can only ever be applied to repeatable experiments like dice or coins, never real-world one-off predictions
    • A 70% probability guarantees it will rain on 70% of the day itself
    • Weather forecasting never actually uses any form of probability calculation

20. 20. Why might simulating a compound event thousands of times using digital tools sometimes reveal an error in a theoretical probability calculation done by hand?

    • If the simulated relative frequency consistently differs from the calculated theoretical probability, it can flag a miscounted outcome or mistaken assumption in the original calculation
    • Simulations and theoretical calculations always match perfectly, so a large mismatch is impossible
    • Simulations can never be used to check the accuracy of a hand calculation
    • A mismatch between simulation and calculation always means the simulation itself is broken

21. 21. Understanding compound probability through trees, tables and simulations mainly helps you to:

    • Systematically calculate and estimate probabilities for events made up of multiple combined outcomes
    • Assume every compound event has an identical probability to a single simple event
    • Avoid using systematic methods like trees or tables to organise outcomes
    • Treat theoretical probability and relative frequency as always giving contradictory results