silemus.
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  1. 1Probability of even on a die
  2. 2P(sum=7)P(\text{sum} = 7) with two dice
  3. 3Sum of two dice ≥10\ge 10
  4. 4Exactly 2 heads in 3 tosses
  5. 5Binomial probability P(X=5)P(X = 5)
  6. 6Binomial P(X=0)P(X = 0)
  7. 7Binomial P(X≥1)P(X \ge 1)
  8. 8Mean and variance, Binomial
  9. 9Poisson P(X=0)P(X = 0)
  10. 10Poisson probability P(X=2)P(X = 2)
  11. 11Poisson P(X≤1)P(X \le 1)
  12. 12Expected value, geometric distribution
  13. 13Choosing 3 from 10
  14. 14Arrangements with repeats
  15. 15Choosing 2 women, 1 man
  16. 16Two cards both aces
  17. 17At least one six in 4 rolls
  18. 18Birthday paradox, 23 people
  19. 19Conditional probability P(A∣B)P(A \mid B)
  20. 20Union of independent events
  21. 21Bayes' theorem on disease test
  22. 22Bayes' theorem on urns
  23. 23Total probability, defective part
  24. 24Expected value of a die
  25. 25Variance of a die roll
  26. 26Expected value of XX
  27. 27Variance of a discrete XX
  28. 28Linear transform of E[X]E[X] and Var(X)\mathrm{Var}(X)
  29. 29Variance of independent sum
  30. 30Uniform probability P(X>7)P(X > 7)
  31. 31Uniform mean and variance
  32. 32Exponential tail probability
  33. 33Exponential wait probability
  34. 34Finding cc for a density
  35. 35Expected value of XX
  36. 36Probability from a density
  37. 37Normal probability, zz-score
  38. 38Normal probability, 40<X<6040 < X < 60
  39. 39zz-score of a normal value
  40. 40Standard error of the mean
  41. 41Sampling distribution of mean
  42. 4295% CI for the mean
  43. 43Sample mean and variance
  44. 44Median of a data set
  45. 45Covariance from E[XY]
  46. 46Correlation from covariance
  47. 47Independence of two events
  48. 48zz test statistic
  49. 49Expected heads in 50 tosses
  50. 50Stationary distribution, 2 states
  51. 51Hypergeometric probability
  52. 52Bayes with a probability tree

12Expected value, geometric distribution

Each trial succeeds with probability 0.2, independently. What is the expected number of trials up to and including the first success?

For trials repeated until the first success, with success probability p, the expected number of trials is 1/p. Here p = 0.2, so E[X] = 1/0.2 = 5 trials, matching the given rate since 1/5 = 0.2.

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