Lecture Notes: Binomial Distribution

Introduction

• Binomial Experiment: A probability experiment satisfying four requirements:
  1. Fixed number of trials (e.g., tossing a coin five times).
  2. Two possible outcomes: success and failure for each trial.
  3. Outcomes are independent of each other.
  4. Probability of success (denoted by ( p )) remains constant for each trial.
• Binomial Distribution: Outcomes and probabilities from a binomial experiment.

Binomial Distribution Formula

• Probability Formula: [ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} ]
  • ( n ): number of trials
  • ( x ): number of successes
  • ( p ): probability of success per trial
  • ( q = 1-p ): probability of failure per trial

Calculating Mean, Variance, and Standard Deviation

• Mean: ( \mu = n \times p )
• Variance: ( \sigma^2 = n \times p \times q )
• Standard Deviation (SD): ( \sigma = \sqrt{n \times p \times q} )

Examples

Example 1: Coin Toss

• Tossing a coin 5 times, probability of exactly 2 tails:
  • ( n = 5 ), ( x = 2 ), ( p = 1/2 )
  • Calculation: ( P(X=2) = \binom{5}{2} (0.5)^2 (0.5)^3 = 0.3125 )
• Mean (expected value): ( 2.5 )
• Variance: ( 1.25 )
• Standard Deviation: ( 1.118 )

Example 2: Political Party Support

• 37% of a community favors a political party; sample of 30 inhabitants:
  • Probability of none voting for the party: ( P(X=0) = 0.0000955 )
  • Probability that exactly 2 vote for the party: ( 0.00014 )
  • Probability that at most 2 vote for the party: ( 0.00016 )
  • Probability that at least 3 vote for the party: ( 0.99984 )
• Mean: ( 11.1 )
• Variance: ( 6.993 )
• Standard Deviation: ( 2.644 )

Example 3: Multiple Choice Test

• 15 questions, each with 5 possible answers, student guessing:
  • Probability of answering at most 3 questions correctly: ( 0.64816 )
• Mean: ( 3 )
• Variance: ( 2.4 )
• Standard Deviation: ( 1.5492 )

Example 4: Traffic Fatalities

• 70% involve an intoxicated driver; sample of 15:
  • Probability exactly 12 involve intoxicated driver: ( 0.17004 )
• Mean: ( 10.5 )
• Variance: ( 3.15 )
• Standard Deviation: ( 1.775 )

Example 5: Dice Roll

• Die rolled 480 times:
  • Mean number of threes: ( 80 )
  • Variance: ( 66.667 )
  • Standard Deviation: ( 8.165 )

Example 6: Product Defects

• Defective rate 3%, sample of 20:
  • Probability of exactly 3 defective items: ( 0.01834 )
  • Probability not more than 2 defective items: ( 0.97899 )
• Mean: ( 0.6 )
• Variance: ( 0.582 )
• Standard Deviation: ( 0.763 )

Example 7: Machine Defects

• Probability of defect 0.1, sample of 3:
  • Probability at most 1 defective: ( 0.972 )
• Mean: ( 0.3 )
• Variance: ( 0.27 )
• Standard Deviation: ( 0.52 )

Example 8: College Graduates

• Probability of graduation 0.6, 3 students:
  • Probability at most 2 students graduate: ( 0.784 )
• Mean: ( 1.8 )
• Variance: ( 0.72 )
• Standard Deviation: ( 0.849 )

Conclusion

• Review concepts of binomial experiment and distribution.
• Practice with examples for better understanding.
• Next topic: Poisson distribution.

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Note: Try additional problems to ensure understanding of the binomial distribution concept and solving methods.