Binomial Distribution

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Bernoulli Trials are independent trials of a random experiment that satisfy two conditions: (i) Each trial has exactly two outcomes: success or failure (ii) The probability of success remains the same

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Yes, these are Bernoulli trials because: (i) Each trial has exactly two outcomes: Even number (success) or Odd number (failure) (ii) Probability remains constant for all trials Probability of success

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Binomial Distribution is the probability distribution of the number of successes in n independent Bernoulli trials. Parameters: • n = number of trials (fixed) • p = probability of success in each tri

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P(X = x) = ⁿCₓ × pˣ × qⁿ⁻ˣ Where: • P(X = x) = Probability of getting exactly x successes • ⁿCₓ = ⁿ!/(x!(n-x)!) = Number of ways to choose x successes from n trials • pˣ = Probability of x successes

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Given: n = 4, p = 1/2 (probability of head), q = 1/2 X ~ B(4, 1/2) Step 1: Identify values x = 2 (exactly 2 heads) Step 2: Apply formula P(X = 2) = ⁴C₂ × (1/2)² × (1/2)² Step 3: Calculate ⁴C₂ ⁴C₂ =

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The probabilities in Binomial Distribution correspond to the terms in the binomial expansion of (q + p)ⁿ. For n = 3: (q + p)³ = q³ + 3q²p + 3qp² + p³ Probability Distribution: • P(X = 0) = q³ (1st t

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Given: n = 5, p = 13/52 = 1/4, q = 3/4, x = 3 X ~ B(5, 1/4) Step 1: Apply formula P(X = 3) = ⁵C₃ × (1/4)³ × (3/4)² Step 2: Calculate ⁵C₃ ⁵C₃ = 5!/(3!×2!) = (5×4)/(2×1) = 10 Step 3: Calculate powers

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For X ~ B(n, p): **Mean (Expected Value):** E(X) = μ = np **Variance:** Var(X) = npq = np(1-p) **Standard Deviation:** SD(X) = σ = √Var(X) = √(npq) Where: • n = number of trials • p = probability