Calculate binomial probabilities, cumulative probabilities, and statistics
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The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials.
Where C(n,k) = n! / (k!(n-k)!) is the binomial coefficient
Example 1: Flip a fair coin 10 times. What's the probability of getting exactly 6 heads?
Solution: n=10, p=0.5, k=6 → P(X=6) = C(10,6) × 0.5^6 × 0.5^4 = 0.2051
Example 2: 20% of students prefer online learning. In a class of 30, what's the probability that at most 5 prefer online?
Solution: n=30, p=0.2, P(X≤5) = cumulative probability
When n is large and p is not too close to 0 or 1, the binomial distribution can be approximated by a normal distribution:
Rule of thumb: Use when np ≥ 5 and n(1-p) ≥ 5
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Use when there are n independent trials, each with two outcomes (success/failure) and constant success probability p.
Mean = n·p, variance = n·p·(1−p), SD = √(n·p·(1−p)).
When n·p ≥ 10 and n·(1−p) ≥ 10, a normal approximation with continuity correction is often reasonable.
PMF gives P(X=k); CDF gives cumulative P(X≤k). Use CDF for ranges and tail probabilities.