courses › Probability & Statistics

Average outcomes, spread, and the binomial count.

Pen and paper is fine · a calculator helps on some problems why?

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Opens at level 33.

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Builds on: Conditional probability (not open yet)

skills · practice stats

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

• Expected value par 25 s not tried yet

  worked example

  X is 76 with probability 0.25 and 19 with probability 0.75. What is the expected value of X?

  Answer: 33.25

  1. Weight each value by its chance: 76 × 0.25 + 19 × 0.75
  2. = 19 + 14.25 = 33.25.

• Variance par 45 s not tried yet

  worked example

  X is 9 with probability 0.3 and 19 with probability 0.7. What is the variance of X?

  Answer: 21

  1. E[X] = 9 × 0.3 + 19 × 0.7 = 16.
  2. E[X²] = 81 × 0.3 + 361 × 0.7 = 277.
  3. Var(X) = E[X²] − (E[X])² = 277 − 256 = 21.

• Exactly k successes par 40 s not tried yet

  worked example

  You flip a fair coin 6 times. What is the probability of getting exactly 1 head? Give the answer as a fraction in lowest terms.

  Answer: 3/32

  1. There are C(6, 1) = 6 orders with exactly 1 head, each with chance (1/2)⁶ = 1/64.
  2. So the probability is 6/64 = 3/32.

• Binomial mean & SD par 20 s not tried yet

  worked example

  Each item a machine makes has a 25% chance of being defective, independently of the others. A batch has 90 items. What is the standard deviation of the number of defective items in the batch? Round to 2 decimal places.

  Answer: 4.11

  1. SD = √(np(1 − p)) = √(90 × 0.25 × 0.75) = √16.875 ≈ 4.1079, which rounds to 4.11.
  2. (The mean is np = 22.5.)

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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