Understanding Chance: Sampling Data

Data Story Examples

  • OJ Simpson: Acquitted in 1995 for the murders of his former wife Nicole Simpson and her friend Ron Goldman but found liable in civil court in 1997. Illustrates legal outcomes diverging based on types of evidence and standards of proof.

  • Sally Clark: Wrongfully convicted of murdering her two sons; conviction overturned after evidence of natural causes was withheld. Highlights the dangers of ignoring or misinterpreting statistical evidence.

Part 1: Chance

  • Prosecutor's Fallacy: Mistake in statistical thinking, assuming the probability of a random match equals the probability of the defendant's innocence. This often leads to overstating the significance of forensic evidence.

    • Example: DNA match of 1 in 500,000 in a city of 5 million.

    • P(DNAMatch∣Innocent)=94,999,999=0.00018%P(DNA Match | Innocent) = \frac{9}{4,999,999} = 0.00018\% (tiny). This is the chance of finding that specific DNA match if the person is innocent.

    • P(Innocent∣DNAMatch)=910=90%P(Innocent | DNA Match) = \frac{9}{10} = 90\% (high). This shows the probability that, given the DNA match, the suspect is still innocent, considering the population size.

  • Properties of Chance:

    • Ranges between 0% (impossible) and 100% (certain).

      • P(Impossible event)=0P(Impossible \ event) = 0

      • P(Certain event)=1P(Certain \ event) = 1

    • The chance of an event equals 100% minus its complement. Understanding complements clarifies probabilities.

      • P(Event)=1−P(Complement event)P(Event) = 1 - P(Complement \ event)

    • Drawing at random gives all objects an equal chance of being picked. This principle is fundamental in ensuring fairness in sampling and probability experiments.

  • Conditional Probability: Chance of an event occurring given another has occurred. It allows for updating probabilities based on new information.

    • P(Event 1∣Event 2)P(Event \ 1 | Event \ 2)

    • Multiplication Principle: P(Event 1 and Event 2)=P(Event 1)×P(Event 2∣Event 1)P(Event \ 1 \ and \ Event \ 2) = P(Event \ 1) \times P(Event \ 2 | Event \ 1) . Key for calculating probabilities of sequential events.

  • Independence: Two events are independent if the chance of the second event is the same regardless of the outcome of the first event. Identifying independence simplifies probability calculations.

    • P(Event 2∣Event 1)=P(Event 2)P(Event \ 2 | Event \ 1) = P(Event \ 2)

    • P(Event \ 1 & Event \ 2) = P(Event \ 1) \times P(Event \ 2)

  • Dependence: Two events are dependent if the chance of the second event depends on the outcome of the first event. Recognizing dependence is crucial for accurate probability assessment.

    • P(2nd event∣1st event)≠P(2nd event)P(2nd \ event | 1st \ event) \neq P(2nd \ event)

  • Mutually Exclusive: Occurrence of one event prevents the other. This simplifies probability calculations because both events cannot occur simultaneously.

    • If Event 1 occurs P(Event 2)=0P(Event \ 2) = 0

  • Addition Rule: If two events are mutually exclusive, the chance of at least one occurring is the sum of their individual chances. Useful in scenarios where only one of several outcomes can occur.

    • P(At least 1 of 2 events)=P(Event 1)+P(Event 2)P(At \ least \ 1 \ of \ 2 \ events) = P(Event \ 1) + P(Event \ 2)

Why Chevalier de Mere Lost Money:

  • Game A: Toss a die 4 times, win if at least one ace (1) is rolled.

  • Game B: Toss two dice 24 times, win if at least one double-ace (two 1s) is rolled.

  • Game A P(Win)=1−(56)4=0.518P(Win) = 1 - (\frac{5}{6})^4 = 0.518 (approx. 52% chance of winning)

  • Game B P(Win)=1−(3536)24=0.49P(Win) = 1 - (\frac{35}{36})^{24} = 0.49 (approx. 49% chance of winning)

Part 2: Binomial Formula

  • Binomial Coefficients:

    • Factorial: n!=n×(n−1)×(n−2)…×3×2×1n! = n \times (n - 1) \times (n - 2) … \times 3 \times 2 \times 1 . The product of all positive integers up to nn.

    • 0!=10! = 1

    • Binomial coefficient: (nx)=n!x!(n−x)!\binom{n}{x} = \frac{n!}{x!(n-x)!} . Represents the number of ways to choose xx items from nn without regard to order.

  • Binomial Theorem:

    • Binary trials: P(event)=p,P(not event)=1−pP(event) = p, P(not \ event) = 1 - p. Describes events with only two possible outcomes: success or failure.

    • Chance of exactly x events occurring in n independent trials: (nx)px(1−p)n−x\binom{n}{x} p^x (1 - p)^{n-x}. Fundamental formula for calculating probabilities in binomial experiments.

    • Notation: X∼Bin(n,p)X \sim Bin(n, p). Shorthand notation indicating that XX follows a binomial distribution with nn trials and success probability pp.