Independent product sample spaces
List all ordered continuations, then use the probability model to weight them
- Build every ordered outcome Fix a result at stage one and list every continuation before changing that first result
- Count combinations Multiply the number of available choices at successive independent stages \(\displaystyle N=n_1n_2\cdots n_k\)
- Check probability assumptions Independent does not mean equally likely. Use favourable counts only when every complete outcome has equal probability
- Weight paths Multiply the stage probabilities along each ordered outcome, then add disjoint favourable paths
- Extend to three stages Retain all three entries. Different orders remain distinct
- Keep model scope clear These product experiments have independent stages. Drawing without replacement requires changed later probabilities, developed with trees next
Worked example
Two independent spinners each show red with probability three quarters and blue with probability one quarter. Find the probability of exactly one red
- \(\displaystyle \Pr(RB)=\frac34\times\frac14=\frac{3}{16}\)First red, then blue
- \(\displaystyle \Pr(BR)=\frac14\times\frac34=\frac{3}{16}\)First blue, then red is a separate outcome
- \(\displaystyle \Pr(E)=\frac{3}{16}+\frac{3}{16}=\frac38\)Add the two disjoint favourable path probabilities
Common mistakes
Difficulty levels
Questions about this topic
Does independent mean equally likely?+
No. An outcome at one stage does not change another stage probability, but different outcomes may still have different probabilities
Why multiply along a path?+
For independent stages, each probability applies within every continuation of the earlier stage
Why add different paths?+
Distinct complete ordered outcomes cannot happen together in the same trial, so they are disjoint
Can I use a table for three stages?+
Yes. Use one column per stage and one row per full triple, or organise slices by a fixed first result
Do lists need a particular order?+
Any systematic order is valid if every possible ordered outcome appears exactly once
What changes without replacement?+
The later available objects and probabilities depend on earlier draws. Keep the order and update the remaining state; the next topic develops this with trees
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 4: Probability