Probability (intro)
The idea
Probability puts a number on chance, on a scale from 0 to 1: impossible events score 0, certain events score 1, and a fair coin's heads sits at 1/2. When every outcome is equally likely, the recipe comes straight from the fractions you know: count the outcomes you want, divide by all outcomes possible. Drawing from a bag of 10 marbles where 3 are blue gives a 3/10 chance of blue, and a useful companion fact says the chance of NOT blue is 1 − 3/10 = 7/10.
The honest interpretation is long-run frequency: a 3/10 probability means that across many, many draws, about 3 in every 10 will be blue — it promises a trend, not a schedule. That kills two popular myths at once. A short run can easily stray from the math, so 40 draws might yield 9 or 15 blues rather than exactly 12; and past results do not bend future chances, so five reds in a row leaves the next draw's probabilities exactly where they started.
Worked example
A bag holds 5 red, 3 blue, and 2 green marbles. You draw one marble at random. Find the probability of blue and of not-green, and estimate how many blues to expect in 40 draws if the marble is replaced each time.
- Count the total outcomes: 5 + 3 + 2 = 10 marbles, each equally likely to be drawn.
- Probability of blue is favorable over total: 3/10, since 3 of the 10 marbles are blue.
- For not-green, subtract from certainty: 1 − 2/10 = 8/10 = 4/5, the same as counting the 5 red plus 3 blue directly.
- For 40 replaced draws, scale the probability up: 3/10 × 40 = 12 blues expected.
- Interpret the expectation honestly: 12 is the long-run typical count, so an actual session might give 10 or 14 blues without anything being wrong with the bag or the math.
Answer. P(blue) = 3/10, P(not green) = 4/5, and you should expect around 12 blue draws out of 40 — as a tendency, not a guarantee.
Check your understanding
- Why must every probability land between 0 and 1, and what would a value outside that range claim?
- How does replacing the marble after each draw keep the probabilities the same, and what changes if you stop replacing it?
- Why does getting five reds in a row not make blue any more likely on the next draw?
- How would you use the not-rule to simplify finding the chance of at least one success in a situation you invent?
Build the foundations first
Probability (intro) builds on these concepts. If any feel shaky, start there.