Key Concepts in Probability

1. Experimental Probability

• Definition: Probability determined based on the results of an experiment.
• Formula: ( P(A) = \frac{N(A)}{N(S)} )
  • ( N(A) ) = number of times event A occurred.
  • ( N(S) ) = total number of trials.
• Example: For a coin flipped 10 times, if tails occurred 3 times, experimental probability ( P(Tails) = \frac{3}{10} ) = 0.3 or 30%.
• More trials lead to closer approximation to theoretical probability.

2. Theoretical Probability

• Definition: Uses reason or math, rather than experimentation, to calculate likelihood.
• Formula: ( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}} )
• Examples:
  • Rolling a die: Probability of rolling a 4 is ( \frac{1}{6} ).
  • Drawing a card: Probability of drawing a seven of hearts is ( \frac{1}{52} ).
  • Picking a marble: Probability of drawing a red marble from 10 marbles (3 red) is ( \frac{3}{10} ).

3. Probability Using Sets

• Intersection of Sets: ( A \cap B ) - common elements in both sets.
  • Example: Probability of liking both hockey and basketball.
• Union of Sets: ( A \cup B ) - elements in either set.
  • Example: Probability of liking basketball or soccer.

4. Conditional Probability

• Definition: Probability of an event given that another event has already occurred.
• Formula: ( P(B|A) = \frac{P(A \cap B)}{P(A)} )
• Example: Probability of liking school given the respondent is female.

5. Multiplication Law

• Independent Events: ( P(A \text{ and } B) = P(A) \times P(B) )
  • Example: Probability of rolling a 3 and flipping tails.
• Dependent Events: ( P(A \text{ and } B) = P(A) \times P(B|A) )
  • Example: Probability of drawing two kings in a row without replacement.

6. Permutations

• Definition: Ordered arrangements of objects.
• Formula: ( n! ) (factorial of n)
• Example: Different orders of letters A, B, C (3! = 6).

7. Combinations

• Definition: Selections where order does not matter.
• Formula: ( C(n, r) = \frac{n!}{r!(n-r)!} )
• Example: Number of groups of 3 from 5 people.

8. Continuous Probability Distributions

• Normal Distribution: Defined by its mean and standard deviation.
• Properties:
  • Mean is center peak.
  • 68% of data within 1 sd, 95% within 2 sd, 99.7% within 3 sd.
• Standard Normal Distribution: Mean = 0, sd = 1.
• Z-Scores: Measure number of standard deviations away from mean.

9. Binomial Probability Distributions

• Definition: Describes probability of k successes in n trials.
• Formula: ( P(X = k) = C(n, k) , p^k , (1-p)^{n-k} )
• Example: Probability distribution of rolling a die multiple times.

10. Geometric Probability Distribution

• Definition: Models number of trials until first success.
• Formula: ( P(X = k) = (1-p)^{k-1} , p )
• Example: Probability that it takes 4 rolls to get doubles.

This summary encapsulates the essence of fundamental probability concepts and is a useful reference for understanding different probability models.