Probability Distributions — Flashcards
Maharashtra Board · Class 12 · Mathematics & Statistics -Commerce
20 flashcards for Probability Distributions (Maharashtra Board Class 12 Mathematics & Statistics -Commerce) to test yourself on key terms and facts.
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What are Bernoulli Trials? State the two conditions that must be satisfied.
Answer
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…
A die is thrown 5 times. If getting an even number is success, are these Bernoulli trials? Explain with probability values.
Answer
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…
Define Binomial Distribution. What are its parameters and notation?
Answer
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…
Write the complete probability function formula for Binomial Distribution and explain each term.
Answer
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 …
A fair coin is tossed 4 times. Find P(X = 2) where X is the number of heads. Show complete step-by-step solution.
Answer
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 =…
What is the relationship between Binomial Distribution and Binomial Expansion? Explain with an example.
Answer
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…
Cards are drawn with replacement from a deck. If drawing a spade is success, find P(exactly 3 spades in 5 draws). Show complete calculation.
Answer
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…
State the formulas for Mean, Variance, and Standard Deviation of Binomial Distribution X ~ B(n, p).
Answer
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 …
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