Probability - Terms

 Here are 30 terms used in probability, along with explained examples and sample calculations, along with real-life examples where applicable:

1. Experiment:

   • Definition: A process or procedure that leads to the observation of an outcome.
   • Example: Flipping a coin.

2. Outcome:

   • Definition: The result of a single trial of an experiment.
   • Example: Getting Heads when flipping a coin.

3. Sample Space:

   • Definition: The set of all possible outcomes of an experiment.
   • Example: For a six-sided die, the sample space is {1, 2, 3, 4, 5, 6}.

4. Event:

   • Definition: A subset of the sample space, or a specific outcome or a combination of outcomes.
   • Example: Rolling an even number on a six-sided die.

5. Probability:

   • Definition: The likelihood of an event occurring, expressed as a number between 0 and 1.
   • Example: The probability of rolling a 4 on a six-sided die is $\frac{1}{6}$.

6. Complement of an Event:

   • Definition: The set of all outcomes not in the event.
   • Example: The complement of rolling a 4 on a six-sided die is rolling a number other than 4.

7. Intersection of Events:

   • Definition: The event containing outcomes common to two or more events.
   • Example: The intersection of rolling an even number and rolling a number greater than 2 on a six-sided die is rolling a 4.

8. Union of Events:

   • Definition: The event containing outcomes from either or both of two events.
   • Example: The union of rolling an even number and rolling a number greater than 2 is rolling a 2, 4, 5, or 6.

9. Independent Events:

   • Definition: Events for which the occurrence of one does not affect the occurrence of the other.
   • Example: Flipping a coin and rolling a die are typically independent events.

10. Dependent Events:

    • Definition: Events for which the occurrence of one affects the occurrence of the other.
    • Example: Drawing two cards without replacement from a deck.

11. Conditional Probability:

    • Definition: The probability of one event occurring given that another event has occurred.
    • Example: The probability of rolling a 6 given that the number is even.

12. Joint Probability:

    • Definition: The probability of the intersection of two events.
    • Example: The probability of rolling a 4 and getting Heads when flipping a coin.

13. Marginal Probability:

    • Definition: The probability of a single event occurring without consideration of other events.
    • Example: The probability of rolling a 4 on a six-sided die.

14. Mutually Exclusive Events:

    • Definition: Events that cannot occur at the same time.
    • Example: Rolling an odd number and rolling an even number on a six-sided die are mutually exclusive events.

15. Equally Likely Outcomes:

    • Definition: Outcomes that have the same probability of occurring.
    • Example: When rolling a fair six-sided die, each number has an equal probability of $\frac{1}{6}$.

16. Random Variable:

    • Definition: A variable whose values are determined by the outcome of a random experiment.
    • Example: The number obtained by rolling a die.

17. Probability Distribution:

    • Definition: A table or function that describes the likelihood of each value of a random variable.
    • Example: The probability distribution of rolling a fair six-sided die.

18. Expected Value (Mean):

    • Definition: The average value of a random variable, weighted by its probabilities.
    • Example: The expected value of rolling a fair six-sided die is $\frac{1}{6} \times 1 + \frac{1}{6} \times 2 + \ldots + \frac{1}{6} \times 6$.

19. Variance:

    • Definition: A measure of the spread of a probability distribution.
    • Example: The variance of rolling a fair six-sided die.

20. Standard Deviation:

    • Definition: The square root of the variance, representing the average distance from the mean.
    • Example: The standard deviation of rolling a fair six-sided die.

21. Binomial Distribution:

    • Definition: A probability distribution that describes the number of successes in a fixed number of independent trials.
    • Example: The number of Heads in 5 coin flips.

22. Poisson Distribution:

    • Definition: A probability distribution that describes the number of events in a fixed interval of time or space.
    • Example: The number of emails received in an hour.

23. Normal Distribution:

    • Definition: A continuous probability distribution with a bell-shaped curve.
    • Example: The distribution of heights in a population.

24. Z-Score:

    • Definition: A measure of how many standard deviations a data point is from the mean.
    • Example: Calculating the Z-score for a height in a population.

25. Confidence Interval:

    • Definition: A range of values used to estimate the true value of a parameter.
    • Example: A 95% confidence interval for the average height in a population.

26. Hypothesis Testing:

    • Definition: A statistical method used to make inferences about a population parameter based on a sample of data.
    • Example: Testing whether the mean exam score of two groups is significantly different.

27. Null Hypothesis:

    • Definition: A statement that there is no effect or no difference.
    • Example: Assuming there is no difference in test scores between two teaching methods.

28. Alternative Hypothesis:

    • Definition: A statement that there is an effect or a difference.
    • Example: Asserting that one teaching method leads to higher test scores than another.

29. Type I Error:

    • Definition: Incorrectly rejecting a true null hypothesis (false positive).
    • Example: Concluding that a new drug is effective when it is not.

30. Type II Error:

    • Definition: Failing to reject a false null hypothesis (false negative).
    • Example: Concluding that a new drug is not effective when it is.

These terms cover a broad range of probability concepts, and each plays a crucial role in understanding and applying probability theory in various real-world situations.