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Probability

Chance

Probability measures how often something happens in the long run. This topic counts outcomes, conditions on what is already known, turns conditional probabilities around with Bayes' theorem, and says when two events are independent.

Lessons on this topic

A unit square holding a circle of radius 0.40, area 0.5027; 1000 darts so far, 504 inside the circle in the accent colour and the rest in grey. Below, a line traces the share inside against the darts thrown, on ticks at 1, 10, 100 and 1000; it is at 0.504 now, with a dotted line at the area.A unit square holding a circle of radius 0.40, area 0.5027; 1000 darts so far, 504 inside the circle in the accent colour and the rest in grey. Below, a line traces the share inside against the darts thrown, on ticks at 1, 10, 100 and 1000; it is at 0.504 now, with a dotted line at the area.

Beginner · 25 minYou are hereRead

What probability measures

Probability is the share of random darts that land in a shape, in the long run, and three simple rules follow from that.

23 people as dots on a ring, every pair joined by a faint line: 253 lines. Persons 15 and 20 and persons 22 and 23 share a birthday, their dots and line lit. Below, the exact chance of a shared birthday against the number of people, from 0 to 60, with a dashed line at one half; a marker at 23 people reads 0.507.23 people as dots on a ring, every pair joined by a faint line: 253 lines. Persons 15 and 20 and persons 22 and 23 share a birthday, their dots and line lit. Below, the exact chance of a shared birthday against the number of people, from 0 to 60, with a dashed line at one half; a marker at 23 people reads 0.507.

Beginner · 30 minYou are hereRead

Counting

Count the ways to arrange and to choose, and see why 23 people are enough for a shared birthday to be more likely than not.

100 people on a 10 by 10 grid: bike owners are filled dots, city people have a ring; 30 people are both. Given that a person lives in the city, the other 50 are faded. The 50 who are left stand in a block, the 30 who also own a bike first. A bar shows 30 of 50, 0.60, with a dashed mark at 0.40 for everyone.100 people on a 10 by 10 grid: bike owners are filled dots, city people have a ring; 30 people are both. Given that a person lives in the city, the other 50 are faded. The 50 who are left stand in a block, the 30 who also own a bike first. A bar shows 30 of 50, 0.60, with a dashed mark at 0.40 for everyone.

Beginner · 30 minYou are hereRead

Conditional probability

Knowing something shrinks the world to what is left, and the new probability is a share of that smaller world.

A grid of 10 000 people, 100 by 100; 100 ill people (1 %) are drawn in the accent colour, the healthy in grey. The test has flagged 99 ill and 495 healthy people (the healthy ones in the second colour); everyone else is dimmed. A box beside the grid holds the 594 people who test positive, the 99 ill first. A bar under the box shows the ill share of the positives, 0.167.A grid of 10 000 people, 100 by 100; 100 ill people (1 %) are drawn in the accent colour, the healthy in grey. The test has flagged 99 ill and 495 healthy people (the healthy ones in the second colour); everyone else is dimmed. A box beside the grid holds the 594 people who test positive, the 99 ill first. A bar under the box shows the ill share of the positives, 0.167.

Intermediate · 30 minYou are hereRead

Bayes' theorem

Turn “how likely is the evidence if the cause is real” into “how likely is the cause, given the evidence”, counting people first.

A row of coin flips, filled dots for heads and open rings for tails, scrolling past a reading head; 10 000 of 10 000 flips so far. A bar below shows the 318 flips that followed five heads in a row: 157 heads and 161 tails, a share of 0.494, with a dashed mark at one half.A row of coin flips, filled dots for heads and open rings for tails, scrolling past a reading head; 10 000 of 10 000 flips so far. A bar below shows the 318 flips that followed five heads in a row: 157 heads and 161 tails, a share of 0.494, with a dashed mark at one half.

Intermediate · 30 minYou are hereRead

Independence

Two events are independent when learning one tells you nothing about the other, as a coin on a lucky streak shows.

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Signal processing
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