Sample spaces, experiments, and the Law of Large Numbers
What this lesson covers
- Some chance-based situations, such as tossing a non-regular 3D shape, can only be explored through probability experiments.
- Results from sets of repeated trials for the same experiment may vary.
- The Law of Large Numbers states that as the number of trials in a chance experiment increases, the experimental probability will approach the experiment’s theoretical probability.
- Lists, tables, two-way tables, and tree diagrams are useful systematic methods for generating all possible outcomes.
Two ways to find a probability
- You can work it out on paper, or you can go and measure it.
- Which one is available depends on the situation.
- And it is not the one you would guess from school examples.
Theory needs something to be symmetrical
- A die has six identical faces, so each is equally likely.
- A drawing pin has nothing like that. No equal faces at all.
- So for the pin there is nothing to calculate. You have to throw it.
| Route | Method | When it works |
|---|---|---|
| Theory | list the outcomes and reason | needs equally likely outcomes |
| Experiment | do it many times and count | works on anything |
| A die | six equal faces | theory works |
| A drawing pin | no symmetry at all | only experiment works |
Which makes school examples misleading
- Dice, coins and cards are almost all you ever meet.
- All three are symmetrical, so theory always works.
- In the real world it usually does not. Experiment is the normal case.
The sample space is every possible outcome
- One die: six outcomes. Two coins: four.
- Theoretical probability is the ways it can happen, over the total.
- So listing the outcomes correctly is the whole job.
| Situation | Outcomes | Count |
|---|---|---|
| One die | 1, 2, 3, 4, 5, 6 | 6 outcomes |
| Two coins | HH, HT, TH, TT | 4 outcomes |
| Theoretical probability | the ways it can happen | over the total |
Two coins, and a question people get wrong
- Toss two coins. What is the chance of one head and one tail?
- Most people say a third: two heads, two tails, or one of each.
- But one of each happens two ways. HT and TH. So it is a half.
| Step | Detail | Value |
|---|---|---|
| Outcomes | HH, HT, TH, TT | |
| One of each | HT and TH | 2 of the 4 |
| So the chance | 2/4 | 1/2 |
| Not 1/3 | HT and TH are different outcomes |
Two dice: use a grid
- Six by six, first die down the side, second along the top.
- Thirty-six cells, and you cannot accidentally miss one.
- Count the cells totalling seven: six of them. So one sixth.
| Step | Working | Result |
|---|---|---|
| Six by six grid | first die down, second across | 36 outcomes |
| Total of 7 | 1+6, 2+5, 3+4, 4+3, 5+2, 6+1 | 6 ways |
| So P(7) | 6/36 | 1/6 |
| Total of 2 | only 1+1 | 1/36 |
Systematic beats careful
- Writing outcomes as they occur to you loses some. Always.
- A grid, a table or a tree has a slot for every single one.
- Then you count the slots rather than your own memory.
| Method | What happens | |
|---|---|---|
| Writing them as they occur | you will lose some | |
| A grid, a table, a tree | has a slot for every one | |
| Count the slots | not the ones you thought of |
Experimental probability is just counting
- How many times it happened, divided by how many trials.
- Eighteen heads in thirty tosses is 0.6.
- That is not 0.5, and nothing has gone wrong.
| Idea | Working | Value |
|---|---|---|
| Definition | times it happened | ÷ number of trials |
| 18 heads in 30 tosses | 18/30 | 0.6 |
| That is not 0.5 | and nothing is wrong |
The pin, where there is no theory to check against
- Throw a drawing pin. Does it land point up or point down?
- There is no symmetry argument. Nothing to calculate.
- Sixty-two point-up in two hundred throws. So about 0.31.
| Step | Detail | Value |
|---|---|---|
| Throw a drawing pin | point up, or point down? | |
| Theory | no equal faces to appeal to | nothing to calculate |
| 62 point-up in 200 | 62/200 | 0.31 |
| That estimate | is the only answer there is |
And that estimate is the answer
- Not an approximation to some true value we could look up.
- There is no formula for a drawing pin.
- Experiment is not second best here. It is the only method.
More trials, closer to the theory
- Thirty tosses gave 0.6, which is off by a tenth.
- A hundred gave 0.56. A thousand gave 0.508.
- The gap keeps shrinking. That is the Law of Large Numbers.
| Trials | Heads | Proportion |
|---|---|---|
| 30 tosses | 18 heads | 0.6, off by 0.1 |
| 100 tosses | 56 heads | 0.56, off by 0.06 |
| 1000 tosses | 508 heads | 0.508, off by 0.008 |
| The gap | shrinks as trials rise | that is the Law |
But it does not mean anything is ever due
- Five heads in a row. Is a tail more likely now? No.
- The coin has no memory. The next toss is still a half.
- What settles down is the proportion, not the counts.
| Claim | Detail | Verdict |
|---|---|---|
| Five heads in a row | is a tail now due? | no |
| The coin | has no memory of the five | |
| Next toss | still 1/2 | exactly as before |
| What converges | the proportion, not the counts |
So how does it settle down, then?
- Those five extra heads never get cancelled out by anything.
- They just stop mattering, once there are a thousand tosses.
- Five out of a thousand is nothing. That is dilution, not correction.
Which is why simulations are worth using
- Nobody is tossing a coin ten thousand times by hand.
- A spreadsheet will do it instantly, as many times as you like.
- And that is when the convergence becomes obvious.
What to take away
- Six things, and the last one wins arguments.
| Theoretical probability | needs outcomes you can list and call equal |
| Experimental probability | is successes divided by trials |
| Some situations | have no theory available at all |
| List outcomes systematically | a grid or a tree, never from memory |
| More trials | brings the experimental value closer |
| Nothing is ever due | the proportion settles, the counts do not |
Your turn
- Pause here and try all four.
- Answers are in the description.
| Question | |
|---|---|
| 1 | How many outcomes when you roll two dice? |
| 2 | What is the probability the two dice total 5? |
| 3 | 24 sixes in 120 rolls. What is P(six)? |
| 4 | After five heads, what is P(head) next? |
Show answer
1. 36. Six faces on the first die times six on the second. 2. 4 out of 36, which is 1/9. The four ways are 1 and 4, 2 and 3, 3 and 2, and 4 and 1. 3. 24 divided by 120, which is 0.2. The theoretical value is one sixth, about 0.167, so this run came out a bit high. More rolls would close the gap. 4. One half. The coin does not know what it did before, and nothing is ever due.
The answers
- Six faces times six faces.
- A total of five happens four ways.
- Nothing is ever due.
| Answer | |
|---|---|
| 1 | 36 |
| 2 | 4/36, or 1/9 |
| 3 | 0.2 |
| 4 | One half |
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