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2.4.3 Examples of Binomial Distributions

Example 1: Biased Die

  • Context: A biased die where the probability of rolling a 5 or 6 (success) is image

  • Binomial Scenario: 10 rolls of the die, counting how many times a 5 or 6 is rolled.

  • Calculations:

  • Expected Number of Successes: image

  • Standard Deviation: image

  • Probability of at Least 3 Successes: Calculated using binomial probability functions.

Example 2: Insurance Calls

  • Context: An insurance agent makes calls, with each call having a 20% chance of leading to a sale (success).
  • Binomial Scenario: 12 calls made per day.
  • Questions:

  • Expected Number of Sales: image

  • Probability of at Least 10 Sales: Computed using binomial probabilities.

Example 3: Flight Delays

  • Context: Observing flight delays at an airport, with a day considered successful if there are two or fewer delays.
  • Binomial Scenario: 30 days observed.
  • Calculations:
  • Probability of at Least 18 Successful Days: Using the binomial distribution for n=30 and p (probability of a successful day).
  • Probability of Between 15 and 25 Successful Days: Summing probabilities from 15 to 25.

Example 4: Defective Pumps

  • Context: Monitoring production quality over several days, defining a successful day as one with one or fewer defective pumps.
  • Binomial Scenario: 14 days monitored.
  • Questions:
  • Expected Number of Successful Days: E[Y]=14×p (where p is the probability of a successful day).
  • Probability of 10 or More Successful Days: Using cumulative binomial probabilities.

General Methodology

  • Use a Spreadsheet to Calculate Binomial Probabilities Efficiently:
  • Setting Up: Enter n (number of trials) and p (probability of success) into the spreadsheet.
  • Using Binomial Functions: Apply Excel's BINOM.DIST function to compute probabilities for various outcomes.
  • Analysis: Analyze scenarios by varying p and checking effects on probabilities, using either direct calculation or goal-seeking methods in Excel.
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