Sample Spaces

Sample Space

What do you think?

If you have 20 pebbles and select 5 of them (all equal weight), what's the probability?

Try it yourself. Below is a board of pebbles. Each pebble represents a possible outcome. Tap some to select them. Change weights with +/- buttons.

Pebble World

Total weight

Weight(A)

P(A)

0.000

Play with it for a minute. Try making one outcome heavier than the others. Select different combinations.

What you just discovered

You've been calculating probability without any formulas. The probability of your selection is just:

$\Large\frac{\text{total weight of selected pebbles}}{\text{total weight of all pebbles}}$

That's it. Every probability calculation follows this pattern. You pick the outcomes you care about, add up their weights, and divide by the total.

A sample space $S$ is the set of all possible outcomes of a random experiment. Each outcome is a sample point.

The pebble board is a sample space. Each cell is one outcome. When all pebbles have equal weight, probability reduces to counting: favorable outcomes divided by total outcomes.

Back to the dice

This explains the Galileo paradox. With two dice, you're not choosing from 11 possible sums (2 through 12). You're choosing from 36 ordered pairs:

$S = \{(1,1), (1,2), (1,3), \ldots, (6,5), (6,6)\}$

Each pair has weight $\frac{1}{36}$ .

What do you think?

How many ways can you roll a sum of 9 with two dice?

Don't merge outcomes that look similar. (3,6) and (6,3) are different outcomes, even though both sum to 9. Collapsing them breaks the math.

Events

An event is any subset of the sample space. You name what you care about, and it becomes an event.

Event description	The outcomes
Rolling an even number (one die)	{2, 4, 6}
Rolling a total of 7 (two dice)	{(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}
Rolling doubles	{(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}

The probability of an event is the sum of weights of outcomes in that event, divided by the total weight. When outcomes have equal weight, it's just counting.

With one fair die, what's P(rolling a number greater than 4)? (decimal, e.g. 0.42)

With two fair dice, what's P(rolling doubles)? (decimal, e.g. 0.42)

Choosing the right sample space

Much of probability comes down to picking a good sample space. A poor choice makes calculations painful. A good choice makes them obvious.

What do you think?

For two dice, which sample space is better: 11 possible sums {2,3,...,12} or 36 ordered pairs {(1,1),(1,2),...}?

Choose a sample space where outcomes have equal probability whenever possible. Then probability reduces to simple counting.

Test your understanding

You flip a coin twice. How many outcomes are in the sample space? (whole number)

For the coin flips above, what's P(exactly one head)? (decimal, e.g. 0.42)

True or False: Events are subsets of the sample space. (true or false)

What's next

Next, we'll look at when the simple "favorable over total" formula works, and when it doesn't.