What is the difference between mutually exclusive and independent events?

Mutually exclusive: A and B cannot occur simultaneously — P(A∩B) = 0. If one occurs, the other cannot. Independent: Occurrence of A doesn't affect probability of B — P(A∩B) = P(A)·P(B). Important: Mutually exclusive events are NEVER independent (unless one has zero probability), because if A occurs, B is impossible — very much affected by A!

When is the binomial distribution used?

Binomial distribution B(n, p) is used when: (1) There are n fixed trials; (2) Each trial has exactly two outcomes — success (p) and failure (q=1−p); (3) Trials are independent; (4) Probability p remains constant for each trial. Examples: tossing a coin n times, quality inspection of n items, number of heads in n tosses.

What is the difference between P(A|B) and P(B|A)?

P(A|B) = P(A∩B)/P(B) — probability of A given B has occurred. P(B|A) = P(A∩B)/P(A) — probability of B given A has occurred. These are generally different! Example: P(it rains|cloudy) is high, but P(cloudy|it rains) = 1 (it's always cloudy when it rains). Bayes' theorem allows conversion between the two.

Bayes' Theorem and Conditional Probability

Probability • Class 12 Mathematics • NCERT • CBSE

Conditional probability: P(A|B) = P(A∩B)/P(B). Bayes' Theorem: P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ) / [ΣP(Eₖ)P(A|Eₖ)]. Binomial distribution: P(X=r) = ⁿCᵣ pʳ qⁿ⁻ʳ. Mean = np, Variance = npq.

Key Formulas

P(A|B) = P(A∩B)/P(B)
P(A∩B) = P(A)·P(B|A)
Bayes: P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ)/ΣP(Eₖ)P(A|Eₖ)
P(X=r) = ⁿCᵣ·pʳ·qⁿ⁻ʳ (Binomial)
Mean = np; Variance = npq (Binomial)

Frequently Asked Questions

What is the difference between mutually exclusive and independent events?: Mutually exclusive: A and B cannot occur simultaneously — P(A∩B) = 0. If one occurs, the other cannot. Independent: Occurrence of A doesn't affect probability of B — P(A∩B) = P(A)·P(B). Important: Mutually exclusive events are NEVER independent (unless one has zero probability), because if A occurs, B is impossible — very much affected by A!
When is the binomial distribution used?: Binomial distribution B(n, p) is used when: (1) There are n fixed trials; (2) Each trial has exactly two outcomes — success (p) and failure (q=1−p); (3) Trials are independent; (4) Probability p remains constant for each trial. Examples: tossing a coin n times, quality inspection of n items, number of heads in n tosses.
What is the difference between P(A|B) and P(B|A)?: P(A|B) = P(A∩B)/P(B) — probability of A given B has occurred. P(B|A) = P(A∩B)/P(A) — probability of B given A has occurred. These are generally different! Example: P(it rains|cloudy) is high, but P(cloudy|it rains) = 1 (it's always cloudy when it rains). Bayes' theorem allows conversion between the two.

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