Probability (intro) · 跟马丁学编程

How many chances you have

You roll a die. Will a six come up? Maybe. Maths can measure that “maybe” — it is called probability.

The basic formula is one sentence: probability = favourable outcomes / all outcomes. A die has 6 faces (all outcomes), a six is one of them (favourable). The probability of a six is 1/6.

It is written as a fraction, a decimal or percents: 1/6 ≈ 0.17 ≈ 17 %. We will most often write fractions — they are exact.

Probability is always between 0 and 1. Zero = impossible (a seven comes up on a die). One = certain (a number from 1 to 6 comes up). The closer to one, the more likely it happens.

One condition people forget: the formula holds only when all outcomes are equally likely. A fair die yes. A loaded die no. Weather neither — “it rains/it does not rain” are not two equal outcomes.

Come and calculate chances — on dice, beads and one wheel of fortune.

Jonáš and Ema play a board game and Jonáš needs to roll a six. “It has not come up three times, now it has to!” Ema shakes her head: “A die has no memory. Each roll is a new chance 1 in 6 — no matter what was before.” Jonáš rolls. A four. “See. And if you wanted an even number, you have three chances out of six, so a half.” Jonáš counts: “So I will bet on even rather than on a six. That is how you play with maths at your back.”

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Always ask yourself two questions in this order: How many are ALL the outcomes? How many of them are FAVOURABLE? Then just a fraction: favourable on top, all on the bottom.

Calculating probability step by step

Step 1: Count all the outcomes. A die: 6. A coin: 2. A bag with 5 red and 3 blue beads: 8.

Step 2: Count the favourable outcomes — the ones the question asks for. “An even number comes up” on a die: even are 2, 4, 6, so 3 outcomes.

Step 3: Divide. P = favourable/all = 3/6 = 1/2. Cancel if you can.

Step 4: A reason check. Did a number between 0 and 1 come out? More than 1 came out → a mistake, probability above one does not exist.

Writing conversions: 1/2 = 0.5 = 50 %. Fraction → decimal: divide. Decimal → percents: times 100.

The opposite event: the probability that something does NOT happen = 1 minus the probability that it does. A six: 1/6. Not-a-six: 1 − 1/6 = 5/6. Sometimes it is easier to calculate the opposite.

What probability does not say: 1/6 does not mean that in six rolls a six comes up exactly once. It means that in thousands of rolls about a sixth of them will be sixes. In the short run chance does what it wants — in the long run it listens to numbers.

Example 1: beads in a bag

Question: In a bag there are 5 red and 3 blue beads. What is the probability that you blindly pull a blue one?

Method: All outcomes: 5 + 3 = 8 beads.

Favourable: 3 blue.

P = 3/8.

Check: 3/8 = 0.375, it is between 0 and 1. It matches. And the opposite: red has 5/8 — together 3/8 + 5/8 = 8/8 = 1. I will pull something for sure.

Answer: The probability of blue is 3/8 (about 37.5 %).

Example 2: a die and conditions

Question: You roll a die. What is the probability that a) a number bigger than 4 comes up, b) an odd number, c) a number smaller than 7?

Method: All outcomes are always 6.

a) Bigger than 4: five and six, so 2 favourable. P = 2/6 = 1/3.

b) Odd: 1, 3, 5, so 3 favourable. P = 3/6 = 1/2.

c) Smaller than 7: all six numbers. P = 6/6 = 1 — certainty.

Answer: a) 1/3, b) 1/2, c) 1. A certain event has probability exactly 1.

Example 3: is the game fair?

Question: A wheel of fortune has 10 equal fields: 4 blue, 3 green, 2 yellow, 1 red. You win on red. A friend claims: “You have a chance 1 in 4, there are 4 colours after all.” Are they right?

Method: The outcomes are not colours, but FIELDS — there are 10 of them and they are all the same size.

Favourable: 1 red field.

P = 1/10, so 10 %. Not 1/4 = 25 %.

Answer: They are not right. Colours are not equally likely, because they do not have equally many fields. You have to count outcomes that are equal — here fields, not colours. This is the most common cheat in thinking about chance.

⚠️

A die has no memory. After three rolls without a six, a six is not “due” — it still has 1/6, as if you were rolling for the first time. Past rolls do not affect future ones. Who believes the opposite loses in lotteries and games for money. Chance does not know the words “it has to now”.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 开始 with ten. When it goes well, add more.

Beads from a bag: favourable divided by all. Answer with a fraction in the form a/b.

Practise the questions

On paper

Dice, pens and one wheel of fortune of your own. For every problem write all/favourable/fraction.

A die roll: calculate the probability for an even number, for a number divisible by three and for a number smaller than 3. Write each as a fraction and in percents.
In a pencil case there are 6 blue, 3 black and 1 green pen. You pull one blindly. What is the probability of black? And what that you do NOT pull green? (Try through the opposite event.)
Design a wheel of fortune with 12 fields so that a win has probability exactly 1/4. How many fields are winning?
A bag: 4 red and 6 blue beads. A friend claims the chance of red is 4/6. Find the mistake and fix it.

Now you 💪

You always count all outcomes first, then favourable, then a fraction.
You check the result: it must be between 0 and 1.
You can use the opposite event (1 minus P) and explain why a die has no memory.

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Done when: You have calculated probabilities on a die and beads, designed a fair wheel of fortune and fixed the mistake 4/6 (correctly 4/10 = 2/5).

What to take from this lesson

Probability = favourable outcomes / all outcomes.
Always between 0 (impossible) and 1 (certain).
Outcomes must be equally likely — count fields, not colours.
Opposite event: P(does not happen) = 1 − P(happens).
Chance has no memory. Past rolls do not change future ones.