Chance

The Probability Scale #

From impossible to certain, all on one scale.

Every chance sits somewhere between impossible and certain

Nothing is less likely than 0, and nothing is more likely than 1.

A fair coin sits halfway along, at $\frac{1}{2}$1/2. That is an even chance.

Six equal sections and one of them shows a 3 — one win in six, so $\frac{1}{6}$1/6.

Sample Space #

Write them all out and counting becomes safe.

Writing out every outcome is what makes counting them safe

A fair coin has just two outcomes, and they are equally likely.

A fair dice has six. List them all and you can count what you need.

Two dice have 36 outcomes. Written in a grid you can count them, not guess.

Single-Event Probability #

The fraction of outcomes that count as a win.

Probability is the fraction of outcomes that count as a win

Four equal sections, so a, b, c and d all have the same chance. Only a wins.

Count winners over sections: 1 out of 4. That fraction is the probability.

Sections only count as chances if they match. Here d is half the disc, so a is 1 in 6.

Complementary Events #

Not happening is one minus happening.

The chance of something not happening is one minus the chance it does

Say the probability of rain is 1 in 5.

The other 4 parts are the complement — the probability it does not: $\frac{1}{5}+\frac{4}{5}=1$1/5 + 4/5 = 1.

Experimental Probability #

A chance you measure instead of calculate.

Doing the experiment gives a chance you measure rather than calculate

40 throws gave 23 heads and 17 tails, not 20 of each.

That measured $\frac{23}{40}$23/40 is the experimental probability — close to $\frac{1}{2}$1/2, not equal.

The Law of Large Numbers #

The more you try, the closer it settles.

The more trials you run the closer the measured chance gets to the true one

10 throws can land far from half. 1000 throws rarely do.

Keep going and the experimental probability settles onto the true one.

Expected Counts from a Chance #

About how many, out of how many tries.

The expected count is the fair share: the chance times the number of tries

Roll a die 60 times. A probability of $\frac{1}{6}$1/6 gives each face a fair share.

The expected count is trials × probability: $60\times \frac{1}{6}$60 × 1/6 is about 10 sixes.

A real run might show 8 or 13. Chance still swings — expected means about.