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Hong KongSimple and compound probability
Skill 6 reads "Solve and interpret real-world problems involving probability using counting procedures, tables, and tree diagrams". Three named tools and not one formula, which tells you the level being tested: counting outcomes systematically, rather than permutations, combinations or expected value.
The tools named in the rule are the best guide to the difficulty. Counting procedures, tables, tree diagrams. No formula is mentioned, and neither are permutations or combinations.
The basic definition
The probability of an event is the number of favorable outcomes divided by the total number of possible outcomes, when every outcome is equally likely.
A bag with 3 red and 5 blue counters has 8 counters. The probability of drawing red is 3 over 8. Every probability sits between 0 and 1, and the probabilities of all outcomes add to 1, which is a free check on any answer.
And, or, and not
| Situation | What to do | Condition |
|---|---|---|
| Both events happen | Multiply | Watch whether the first draw changes the second |
| Either event happens | Add, then subtract the overlap | Nothing to subtract if they cannot both happen |
| The event does not happen | Subtract from 1 | Always available |
| At least one happens | Take 1 minus the probability of none | Faster than adding cases |
The last row is the most useful line on this page. "At least one" problems are painful to add up case by case and trivial to do backwards.
With and without replacement
This is the distinction that separates a right answer from a plausible wrong one. If the first item is put back, the second draw faces the same pool. If it is not, both the favorable count and the total drop by one.
Read the stem for the words replaced, put back, kept or without replacement. If the stem does not say, look at the situation: eating a sweet is without replacement, rolling a die is with.
A box holds 4 red and 8 blue counters. Two are drawn without replacement. What is the probability that both are red?
- 1 over 9
- 1 over 11
- 1 over 3
- 2 over 12
Tree diagrams and tables
The rule names both, so expect to be given one or to need to sketch one. A tree lists every path through a sequence of events; a table lists every combination of two.
They are worth using rather than avoiding. On a two-stage problem, drawing the tree takes thirty seconds and removes the whole class of errors that comes from multiplying the wrong pair. You have 2.86 minutes a question on this subtest, on our division of the published figures.
What is not here
No skill mentions permutations, combinations, factorials, conditional probability notation or expected value. "Counting procedures" is the rule's phrase and it means systematic listing, which is what a tree diagram is.
If a study product is teaching you the combinations formula for this test, it is preparing you for a harder exam than the one Florida's rule describes.
If you failed Mathematics
Probability is one skill of the seven in Competency 4, and Competency 4 is about 11 of the 35 questions on our split, which is our arithmetic rather than a published weighting. So this is likely one or two items, and it is not where a failure came from.
It is, though, one of the cheapest skills to fix. The with-and-without-replacement distinction and the at-least-one shortcut together cover most of what a question at this level can do, and both can be learned in an evening. Mathematics passed 71% of first-attempt candidates in 2025 and 83% at best attempt, so a retake with real preparation behind it is a reasonable bet at USD 32.50 and thirty days.
Common questions
What probability is on the FTCE General Knowledge Test?
One skill of the seven in the data competency, covering real-world problems solved with counting procedures, tables and tree diagrams. No formula is named, and permutations, combinations and expected value appear nowhere in the twenty-one Mathematics skills.
How do I handle drawing without replacement?
Reduce both counts after the first draw. If four of twelve counters are red, the first draw is four over twelve and the second is three over eleven. Treating the second draw as if the counter had been replaced is the standard wrong answer.
What is the fastest way to do an at-least-one problem?
Work backwards. The probability of at least one occurrence is one minus the probability of none. Adding the cases individually is slower and gives more chances to miss one, and the subtraction is a single step.
Do I need permutations and combinations?
No skill mentions them. The rule says counting procedures, tables and tree diagrams, which means systematic listing rather than formulas. Material teaching the combinations formula for this test is preparing you for a harder exam than Florida’s rule describes.