doing math

courses › Probability & Statistics

Chance basics

level 18 course

Probability as favourable outcomes over all outcomes.

Pen and paper is fine · no calculator needed why?

Learn first (about 3 minutes)

Opens at level 18.

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Builds on: Fraction arithmetic (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

A probability is a share: the outcomes you want over all the equally likely outcomes. A bag of 3 red and 5 blue marbles gives red a chance of 3/8.

The outcomes must be equally likely. Two dice make 36 equally likely pairs, not 11 equally likely sums: six pairs make a sum of 7, and only one makes 2.

For "at least one", use the opposite. Work out the chance it never happens, then take that from 1.

Techniques

Wanted over total

One pick from a bag, or any set of equally likely outcomes.

  1. Count every outcome: all the marbles, not only the other colours.
  2. Count the ones you want. For "not red", count everything that is not red.
  3. Write wanted over total and reduce.
worked example

Example: A bag has 4 red, 6 blue and 2 green marbles. You pick one at random. What is the probability it is not blue? Answer as a fraction.

  1. 4 + 6 + 2 = 12 marbles.
  2. Not blue: 4 + 2 = 6.
  3. 6/12 = 1/2.

Answer: 1/2

Count the 36 pairs

Two dice.

  1. Two dice make 6 × 6 = 36 equally likely pairs.
  2. A sum of 7 has six pairs, and each step away from 7 loses one. Doubles are six pairs.
  3. For "at least", add the pairs for each sum, or subtract the rest from 36.
worked example

Example: You roll two fair six-sided dice. What is the probability that the sum is at least 10? Answer as a fraction.

  1. Sums 10, 11 and 12: 3 + 2 + 1 = 6 pairs.
  2. 6/36 = 1/6.

Answer: 1/6

Use the opposite

The chance of at least one success in several tries.

  1. Find the chance of failing on one try: no 6 on a die is 5/6.
  2. Multiply that chance once for every try.
  3. Subtract the result from 1.
worked example

Example: You flip a fair coin 3 times. What is the probability of getting at least one head? Answer as a fraction.

  1. No heads: 1/2 × 1/2 × 1/2 = 1/8.
  2. At least one head: 1 − 1/8 = 7/8.

Answer: 7/8

Tips by skill

  • TipOne event: Everything in the bag goes on the bottom. For "not", count everything except that colour on top.
  • TipTwo dice: Count pairs out of 36. Sums near 7 have the most pairs; 2 and 12 have one each.
  • TipAt least one: Find the chance it never happens, multiplying once per try, then subtract that from 1.

Watch out for

  • Dividing by the other colours instead of by everything. With 5 blue out of 12, the chance of blue is 5/12, not 5/7.
  • Treating the 11 possible sums of two dice as equally likely.
  • Adding chances for "at least one". Two rolls at 1/6 each do not make 2/6, because that counts a double six twice.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

your rounds

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