Design and interpret a probability simulation

A trial tests whether two drawn counters are both red. What counts as one complete trial?

Run the counter experiment. Start with two draws, compare both replacement rules, then increase the trial count and change the run number. Inspect the pictured individual counters in the last trial. Try three draws too. Success always means every draw in that trial is red. The same bag is restored before each new complete trial

For a two-draw trial without replacement, when is the whole bag restored?

Design, conduct and evaluate

State the event, the complete trial, the random selection method and both reset rules. Match the actual changing state, conduct repeated trials and calculate the success proportion

1. 1Within a trial Follow the target replacement rule between its draws
2. 2Between trials Restore the initial state when repeating the same complete experiment
3. 3Mapping Equal device outcomes must reproduce the intended item probabilities
4. 4Conclusion A simulation result describes its model. It does not prove that model fits a real process

A bag has 3 red and 2 blue counters. A random integer from 1 to 5 is equally likely. Which mapping models a first draw?

Number R1,R2,R3,B1,B2 as 1 to 5. For two draws without replacement, what should happen if the second generated number repeats the first?

Could you instead select uniformly from a list of the four remaining counter IDs?

After a first red draw is kept out, how many red counters remain?

Red remaining

\(\displaystyle 2\)

Blue remaining

\(\displaystyle 2\)

After that first red draw is kept out, what is the probability that the second draw is red?

Red initially

\(\displaystyle 6\)

Blue initially

\(\displaystyle 4\)

Red kept out

\(\displaystyle 1\)

A proposed model doubles the bag to 6 red and 4 blue. After a first red is kept out, what is its next-red probability?

A bag has 3 red and 2 blue counters. A proposed model has 6 red and 4 blue. Does the larger bag model two draws without replacement from the original bag?

For independent draws with replacement, can 6 red and 4 blue model 3 red and 2 blue?

A run of paired draws produces these results. How many complete trials succeeded if success means both red?

Successful pairs

\(\displaystyle 2\)

Complete pairs

\(\displaystyle 5\)

Find the relative frequency of success in that run

A student divides the success count by 10 because each of five trials has two draws. What went wrong?

Two valid runs of the same model give different estimates. What is the best next step?

A large run is close to a known model probability. What does that establish about a real cap-toss experiment?

Starting red

\(\displaystyle 3\)

Starting blue

\(\displaystyle 2\)

Red counters kept out

\(\displaystyle 2\)

For three draws without replacement from 3 red and 2 blue, after two reds have been drawn, what is the third-red probability?

First red branch

\(\displaystyle 3/5\)

Second red after R

\(\displaystyle 2/4\)

Third red after RR

\(\displaystyle 1/3\)

Three-red probability

\(\displaystyle \frac35\times\frac24\times\frac13=\frac1{10}\)

Find the probability of three reds without replacement

Red probability on each draw

\(\displaystyle 3/5\)

Independent draws

\(\displaystyle 3\)

Three-red probability

\(\displaystyle \frac35\times\frac35\times\frac35=\frac{27}{125}\)

Find the probability of three reds with replacement

How should you evaluate the digital investigation in a written report?

The next practice task may define a trial as one selection, with removed items kept out for the whole run. Should you automatically apply the pair laboratory’s reset-after-two-draws rule?