Binomial Distribution Notes

Introduction to the Binomial Distribution

• The binomial distribution is a crucial probability distribution that helps determine the likelihood of successes in a series of independent trials.

Scenario

• Consider a restaurant promotion where each meal purchase comes with a coupon.
• 20% of coupons win a free milkshake, while the rest indicate "better luck next time."
• If 10 people order, what's the probability that exactly 3 win a milkshake?
• Let $x$ represent the number of winners out of the 10 people.
• The goal is to find the probability distribution of $x$.

Definition of Binomial Distribution

• In a series of trials, let $x$ be the number of successes.
• A random variable representing the number of successes in a series of trials has a probability distribution called the binomial distribution.

Conditions for Binomial Distribution

• Fixed Number of Trials: A specific number of trials ($n$) is conducted.
• Two Possible Outcomes: Each trial results in either a success or a failure.
• Constant Probability of Success: The probability of success ($p$) remains the same for each trial.
• Independent Trials: The outcome of one trial does not influence the outcome of others.
• Random Variable x: Represents the number of successes that occur.

Notation

• $n$: Number of trials conducted.
• $p$: Probability of success on each trial.

Examples

Example 1: Coin Toss

• Experiment: A coin is tossed 10 times.
• Variable: $x$ is the number of times the coin lands heads.
• This is a binomial experiment:
  • Each toss is a trial.
  • Outcomes: heads or tails.
  • $x$ represents the number of heads (success).
  • Trials are independent.

Example 2: Basketball Free Throws

• Experiment: Five basketball players each attempt a free throw.
• Variable: $x$ is the number of free throws made.
• This is NOT a binomial experiment:
  • The probability of making a shot differs from player to player.

Example 3: Drawing Cards

• Experiment: 10 cards in a box (5 red, 5 green); three cards drawn at random.
• Variable: $x$ is the number of red cards drawn.
• This is NOT a binomial experiment:
  • Trials are not independent.
  • Drawing a red card on the first trial changes the probability of drawing another red card on subsequent trials.

Calculating Binomial Probabilities with Excel

• Use the BINOM.DIST command in Excel.
• Arguments:
  • Number of successes ($x$).
  • Number of trials ($n$).
  • Probability of success ($p$).
  • Cumulative (TRUE) or exact (FALSE) probability.
    • FALSE: Probability of exactly $x$ successes.
    • TRUE: Probability of $x$ or fewer successes.

Example: Pinterest Users

• 30% of internet users in the U.S. use Pinterest.
• A sample of 15 internet users is taken.

Scenario 1: Probability of Exactly Four Users Using Pinterest

• $x = 4$ (number of successes).
• $n = 15$ (number of trials).
• $p = 0.3$ (probability of success).
• Command: =BINOM.DIST(4, 15, 0.3, FALSE).
• Result: 0.2186.

Scenario 2: Probability of Fewer Than Three Users Using Pinterest

• Fewer than three is equivalent to two or fewer.
• Command: =BINOM.DIST(2, 15, 0.3, TRUE).
• Result: 0.1268.

Creating a Table of Probabilities

• Create a table listing all values of $x$ from 0 to 15.
• Use the BINOM.DIST command with the cell referencing $x$ as the first argument and FALSE as the last argument.
• Copy the formula to all cells in the table.

Scenario 3: Probability of More Than One User Using Pinterest

• Sum all probabilities corresponding to $x$ values greater than 1.
• Result: 0.9647.

Scenario 4: Probability Between One and Four Users (Inclusive)

• Sum all probabilities corresponding to $x$ values between 1 and 4 (inclusive).
• Result: 0.5107.

Mean, Variance, and Standard Deviation of a Binomial Random Variable

• Let $X$ be a binomial random variable with $n$ trials and success probability $p$.

Formulas

• Mean: $\mu_x = n \times p$
• Variance: $\sigma^2_x = n \times p \times (1 - p)$
• Standard Deviation: $\sigma_x = \sqrt{n \times p \times (1 - p)}$

Example: Car Repairs

• The probability that a new car requires repairs during warranty is 0.15.
• A dealership sells 25 such cars.
• Let $X$ be the number of cars that will require repairs during the warranty period.

Calculations

• $n = 25$
• $p = 0.15$
• Mean: $\mu_x = 25 \times 0.15 = 3.75$
• Variance: $\sigma^2_x = 25 \times 0.15 \times (1 - 0.15) = 3.1875$
• Standard Deviation: $\sigma_x = \sqrt{25 \times 0.15 \times (1 - 0.15)} = 1.785$

Coronary Bypass Surgery Example

• 53% of coronary bypass patients are over 65.
• 15 patients are sampled.

Scenario 1: Probability of Exactly Nine Patients Over 65

• $x = 9$
• $n = 15$
• $p = 0.53$
• Command: =BINOM.DIST(9, 15, 0.53, FALSE)
• Result: 0.178

Scenario 2: Probability of More Than 10 Patients Over 65

• More than 10 is the complement of 10 or fewer.
• Command: =1 - BINOM.DIST(10, 15, 0.53, TRUE)
• Result: 0.092

Scenario 3: Probability of Fewer Than Eight Patients Over 65

• Fewer than eight is equivalent to seven or fewer.
• Command: =BINOM.DIST(7, 15, 0.53, TRUE)
• Result: 0.4065