Binomial Distribution

Coin Toss Probability Outcomes

  • Probability Basics

    • When flipping a coin, there are two possible outcomes: heads (H) and tails (T).

    • Each outcome has a fixed probability from trial to trial.

    • For a fair coin:

    • Probability of heads, $P(H) = \frac{1}{2}$

    • Probability of tails, $P(T) = \frac{1}{2}$

    • It is important to note that biased coins also exist, which have unequal probabilities for heads and tails.

  • Binomial Distributions

    • This section focuses on probability distributions with just two possible outcomes, defined as binomial distributions.

Multiple Coin Toss Scenarios

  • Outcomes of Flipping a Coin Twice

    • The four possible outcomes when flipping a coin two times are:

    1. HH (2 heads)

    2. HT (1 head, 1 tail)

    3. TH (1 head, 1 tail)

    4. TT (0 heads)

    • Each outcome occurs with a probability of $P = \frac{1}{4}$ because the tosses are independent.

    • Probability calculation:

      • For HH:

      • $P(HH) = P(H1) \cdot P(H2) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

      • For HT:

      • $P(HT) = P(H1) \cdot P(T2) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

      • The same probability applies to TH.

  • Classification of Outcomes

    • Outcomes can be categorized by the number of heads:

    • 0 Heads: Outcome 4 (TT)

    • 1 Head: Outcomes 2 (HT) and 3 (TH)

    • 2 Heads: Outcome 1 (HH)

    • Consequently, the probabilities are summarized as follows:

    • $P(2) = \frac{1}{4}$

    • $P(1) = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}$

    • $P(0) = \frac{1}{4}$

Binomial Distribution Overview

  • Definition and Parameters

    • A binomial distribution comprises the probabilities of achieving a certain number of successes in a set number of trials (n), each with a fixed success probability (π).

    • Key Variables:

    • $P(x)$ = Probability of x successes in n trials

    • n = Number of trials

    • π = Probability of success on a given trial

  • Example Application

    • For coin flipping where n=2 and π=0.5, the probabilities can be computed using the binomial distribution formula.

Calculating Specific Probabilities

  • Probability of Getting One or More Heads

    • When flipping a coin twice, the probability of:

    • Exactly one head = $P(1) = 0.5$

    • Exactly two heads = $P(2) = 0.25$

    • Therefore, the probability of getting one or more heads is:

    • P(1 ext{ or more heads}) = 0.5 + 0.25 = 0.75

  • Biased Coin Example

    • If the coin is biased with $P(H) = 0.4$, the probability of getting heads at least once in two tosses:

    • Use the binomial formula to find the solution:

      • P(at ext{ least one head}) = 1 - P(0)

      • Results in P(at ext{ least one head}) = 0.64

  • Probability of Zero to Three Heads in Twelve Tosses

    • For a scenario of flipping a coin 12 times:

    • Calculate probabilities for exactly 0, 1, 2, and 3 heads:

      • $P(0) = 0.0002$

      • $P(1) = 0.0029$

      • $P(2) = 0.0161$

      • $P(3) = 0.0537$

    • The sum of these probabilities gives:

      • P(0 ext{ to } 3) = 0.0002 + 0.0029 + 0.0161 + 0.0537 = 0.073

Mean and Variance of Binomial Distribution

  • Calculating the Mean

    • The mean number of heads from 12 coin tosses can be determined:

    • On average, half the tosses yield heads, thus:

      • \mu = n \times \pi = 12 \times 0.5 = 6

  • Calculating the Variance

    • Variance of the binomial distribution is calculated as:

    • \sigma^2 = n \times \pi \times (1 - \pi)

    • In the coin toss example with $n = 12$ and $\pi = 0.5$:

    • \sigma^2 = 12 \times 0.5 \times (1 - 0.5) = 12 \times 0.5 \times 0.5 = 3.0

  • Standard Deviation

    • Standard deviation is defined as:

    • \sigma = \sqrt{\sigma^2}

    • Thus, the standard deviation is found as the square root of the variance.