Probability and Binomial Distribution Lecture

Announcements

• Upcoming holiday on Thursday, 4th of July; no classes.
• Material covered today, tomorrow, and Wednesday will be included in the next exam.

Section 11.4: Binomial Probability

Binomial Probability Distribution

• Berui Trials: Another name for binomial trials, named after a mathematician.
• Success and Failure: For binomial experiments, success and failure are the two outcomes.

Sample Space Example

• Spinner Example: Spinner is spun twice, outcomes labeled.
• Result Interpretation: Success or failure based on where the spinner lands.
• Binomial Distribution Basis: Experiment divided into success and failure.
• Constant Probability: Probability of success remains constant through trials.

Random Variables

• Random Variable vs. Controlled Variable: Random variable's values are determined by random processes.
• Example: Spinning a spinner, controlled by the outcome of the spin.
• Sample Space Analysis: Possible outcomes listed and analyzed.
• Probability Distribution Table: List all possible values of X and their probabilities.

Binomial Probability Formula

• General Formula: [P(X) = \binom{n}{x} p^x (1-p)^{n-x}]
• Definitions: n = number of trials, p = probability of success, q = 1-p, x = number of successes.

Examples

• Coin Toss: Probability of exactly three heads in five tosses.

  • Binomial Experiment: 5 trials, success probability = 0.5, failure = 0.5.
  • Calculation: [P(X=3) = \binom{5}{3} (0.5)^3 (0.5)^2]

• Dice Roll: Probability of exactly two threes in six rolls.

  • Success = rolling a 3, failure = anything else.
  • Calculation: [P(X=2) = \binom{6}{2} (1/6)^2 (5/6)^4]

More Advanced Examples

• Different Outcomes: Probability of no threes in six rolls, all six rolls are threes.
• Deriving Probabilities: Using combinations and probabilities to derive different scenarios.

Expected Value

• Concept: Long-term average of a random variable; weighted average.
• Formula: [E(X) = \sum_{i} x_i * P(x_i)]*_

Practical Examples

• Child Gaming Hours: Survey results showing the distribution of gaming hours.

  • Calculation: Sum of products of hours and their probabilities.

• Fair Game: Game where expected net winnings are zero.

  • Example: Calculating the entry fee to make a betting game fair.
  • House Advantage: Change in net gain perspective for player and game maker.

Roulette Example

• American Roulette: 38 compartments, 18 each red and black, one zero, one double zero.
• Betting on Color: Calculation of expected net winnings when betting on red or black.

Conclusion

• Review of Concepts: Binomial probability, sample space, expected value, and fair game.
• Next Steps: Will start on section 11.5 tomorrow.

Key Takeaway: Probability and expected value calculations are crucial for determining outcomes in various scenarios including games and experiments.