Binomial Distribution Lecture

Definition and Conditions

• Binomial Distribution: A type of probability distribution with specific conditions, summarized by the acronym "BINS."
  • B: Binary outcomes
    • There are only two outcomes in the event: Success or Failure.
  • I: Independent trials
    • Success or failure of one event does not affect another.
  • N: Number of trials
    • There is a defined number of trials, denoted as 'n'.
  • S: Same probability of success
    • Each trial must have the same probability of success.

Example

• Rolling a Die
  • Rolling a die 4 times, trying to roll a '1' twice.
  • Binary: Either roll a '1' (success) or not (failure).
  • Independent: Each roll is independent of the previous.
  • Number of Trials: 4 rolls.
  • Same Probability: Probability of rolling a '1' is 1/6 in each trial.

Properties of Binomial Distribution

• Mean (Expected Value)
  • Formula: ( \mu = n \times p )
  • ( n ) = number of trials, ( p ) = probability of success.
• Standard Deviation
  • Formula: ( \sigma = \sqrt{n \times p \times (1-p)} )
  • ( q = 1-p ) (probability of failure).

Calculating Probability

• Formula: ( P(X = x) = \binom{n}{x} \times p^x \times (1-p)^{n-x} )
  • ( \binom{n}{x} ): "n choose x" - number of ways to achieve x successes in n trials.
  • Example calculation with rolling a die:
    • Setup: Probability of rolling a '1' twice in 4 trials.
    • Values: ( n = 4 ), ( p = 1/6 ), ( x = 2 ).
    • Calculation:
      • Probability = ( \binom{4}{2} \times (1/6)^2 \times (5/6)^{4-2} )
      • ( \binom{4}{2} = 6 )
      • Final probability = 0.1157 or 11.57%

Mean and Standard Deviation for Example

• Mean: ( \mu = 4 \times (1/6) = 2/3 )
  • Interpretation: Expected to roll a '1', 2/3 times in 4 trials.
• Standard Deviation: ( \sigma = \sqrt{4 \times (1/6) \times (5/6)} = 0.745 )

Conclusion

• Understanding binomial distribution involves knowing the conditions, calculating probabilities, and using the formulas for mean and standard deviation.
• Practice with examples, like rolling a die, can help in mastering these concepts.
• Feedback and questions are encouraged to improve learning.