Binomial Distribution 🪙

The Binomial Distribution is a discrete probability distribution used when there are exactly two possible outcomes for each trial: Success or Failure. It was discovered by James Bernoulli (hence also called Bernoulli Distribution).

Conditions (Assumptions) 📝

1. Finite Trials: The number of trials (n) is finite and fixed.
2. Two Outcomes: Every trial results in either Success (p) or Failure (q).
3. Independent Trials: The outcome of one trial does not affect another.
4. Constant Probability: The probability of success (p) remains constant in every trial.

[!NOTE] Relationship: p + q = 1 or q = 1 - p.

The Probability Mass Function (PMF) 📐

The probability of getting exactly r successes in n trials is given by:

P(r) = nCr * p^r * q^(n-r)

Where:

• n = Total number of trials
• r = Number of successes required
• p = Probability of success in one trial
• q = Probability of failure (1-p)
• nCr = Number of ways to select r successes from n trials.

Example

Tossing a coin 3 times. Find probability of getting exactly 2 Heads.

• n = 3
• r = 2
• p = 0.5 (Head)
• q = 0.5 (Tail)

P(2) = 3C2 * (0.5)^2 * (0.5)^(3-2) P(2) = 3 * 0.25 * 0.5 = 0.375

Shape 📉

• Symmetrical: If p = q = 0.5.
• Skewed Right: If p < 0.5.
• Skewed Left: If p > 0.5.

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