Binomial Distribution

Binomial Experiments

• Definition: A binomial experiment consists of a fixed number (n) of trials.

  • Characteristics:

    • Trials are mutually independent.

    • Each trial has only two possible outcomes: success or failure.

Probability of Success and Failure

• Success probability: p

• Failure probability: 1 - p

• Random variable (X): counts total successes out of n trials.

  • Example: Coin flipping.

    • Random variable X may represent the number of heads in 4 flips of a fair coin (n=4, p=0.5).

  • Example: Free throws.

    • X may count successful free throws from a player out of 10 attempts (n=10).

    • p is considered constant across independent attempts.

Binomial Probability Distribution Formula

• Formula: P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)

  • C(n, k) is the number of combinations (n choose k).

  • p^k is the probability of k successes.

  • (1 - p)^(n - k) is the probability of n-k failures.

Example Calculation of Probability

• Consider n = 4 and k = 2 (successes):

  • Combinatorial arrangements: C(4, 2) = 6 (from combinatorics).

  • Probability of specific arrangement: p^2 * (1-p)^2.

Mean and Variance of Binomial Distribution

• Mean ( \mu): n * p

  • Intuition: If flipping 8 coins (n=8, p=0.5), expect 4 heads.

• Variance ( \sigma^2): n * p * (1 - p)

Example Problem

• Dennis's Attendance Probability:

  • Probability of Dennis attending class = 0.8; number of days (n) = 4.

  • Expected value: E(X) = n * p = 4 * 0.8 = 3.2.

  • Variance: Var(X) = n * p * (1 - p) = 4 * 0.8 * 0.2 = 0.64.

Probability of Attending No More Than 2 Classes

• Calculate P(X ≤ 2):

  • P(X = 0) + P(X = 1) + P(X = 2).

  • Using the formula:

    • P(X = 0) = C(4, 0) * (0.8)^0 * (0.2)^4.

    • P(X = 1) = C(4, 1) * (0.8)^1 * (0.2)^3.

    • P(X = 2) = C(4, 2) * (0.8)^2 * (0.2)^2.

  • Total Probability = 0.181.

Computational Tools

• Tools for Calculation:

  • Scientific calculators can compute exact binomial probabilities P(X = k) or cumulative probabilities P(X ≤ k).

  • Microsoft Excel:

    • Function: =BINOM.DIST(k, n, p, TRUE) for cumulative probabilities.

    • Use FALSE for individual probabilities.

Binomial Distribution Table

• Characteristics:

  • Tables represent cumulative probabilities by n and p.

  • Lookup for specific k cutoff values.

• Example for Dennis:

  • For n=4, p=0.8 and k=2, find cumulative probability = 0.1808 in table.

Conclusion

• Understanding binomial experiments is crucial for various applications in statistics and probability theory.