doing math

courses › Probability & Statistics

Expected value & variance

level 33 course

Average outcomes, spread, and the binomial count.

Pen and paper is fine · no calculator needed why?

Learn first (about 3 minutes)

Opens at level 33.

Sign in to start

Builds on: Conditional probability (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 random variable is a number decided by chance, like a game's payout. Its expected value is the long-run average: weight each value by its probability and add.

The variance is the average squared distance from that expected value, and its square root is the standard deviation. A binomial count is the number of successes in n independent tries, each with the same chance p. Its mean is np and its variance is np(1 − p).

Techniques

Weight and add

The expected value or the variance, from values and their probabilities.

  1. Expected value: multiply each value by its probability and add. Keep any minus signs.
  2. Mean of the squares: the same, with each value squared.
  3. Variance: the mean of the squares minus the expected value squared.
worked example

Example: X is 0 with probability 0.8 and 5 with probability 0.2. What is the variance of X?

  1. Expected value: 5 × 0.2 = 1.
  2. Mean of the squares: 25 × 0.2 = 5.
  3. Variance: 5 − 1 × 1 = 4.

Answer: 4

One order, times the orders

The chance of exactly k successes in n tries.

  1. One order: p for each success and 1 − p for each failure, multiplied.
  2. Number of orders: C(n, k), the ways to choose which k tries succeed.
  3. Multiply the two and reduce.
worked example

Example: You flip a fair coin 4 times. What is the probability of exactly 2 heads? Answer as a fraction.

  1. One order, like HHTT: 1/2 × 1/2 × 1/2 × 1/2 = 1/16.
  2. Orders: C(4, 2) = 6.
  3. 6 × 1/16 = 6/16 = 3/8.

Answer: 3/8

Binomial mean and spread

The mean or standard deviation of a count of successes.

  1. Mean: n × p.
  2. Standard deviation: the square root of n × p × (1 − p). Round only at the end.
worked example

Example: An event has a 25% chance on each of 200 independent trials. What is the standard deviation of the number of successes? Round to 2 decimal places.

  1. Variance: 200 × 0.25 × 0.75 = 37.5.
  2. The square root of 37.5 ≈ 6.12.

Answer: 6.12

Tips by skill

  • TipExpected value: Multiply each value by its probability, then add. Keep the minus sign on any negative value.
  • TipVariance: Mean of the squares minus the square of the mean. For two equally likely values, square half the gap.
  • TipExactly k successes: Chance of one order (p per success, 1 − p per failure) times C(n, k), the number of orders.
  • TipBinomial mean & SD: Mean is np. Standard deviation is the square root of np(1 − p). Round only at the end.

Watch out for

  • Averaging the values and ignoring the probabilities.
  • Giving the standard deviation when the question asks for the variance, or the other way round.
  • Finding the chance of one particular order and forgetting the others. Exactly 3 heads in 5 flips can happen in 10 orders.

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

No rounds yet.