Chance

The Probability Scale #

From impossible to certain, all on one scale.

Every chance sits somewhere between impossible and certain

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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

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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

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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

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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

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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

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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

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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.