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

36Probability from a density

X\displaystyle X has density f(x)=2x\displaystyle f(x) = 2x on [0,1]\displaystyle [0, 1]. Find P(X≤0.5)\displaystyle P(X \le 0.5).

Probability is the area under the density curve, so P(X ≤ 0.5) = ∫₀^0.5 2x dx. The antiderivative of 2x is x², giving (0.5)² − 0² = 0.25. Check: the full area ∫₀¹ 2x dx = 1, confirming f(x) is a valid density.

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