Binomial Distribution Overview
A discrete random variable \( X \) is said to follow a binomial distribution with parameters \( n \) and \( p \) if it takes on values \( x = 0, 1, 2, …, n \) with the probability mass function:
\[
P(X = x) = \binom{n}{x} p^x q^{n – x}, \quad \text{where } q = 1 – p
\]
- \( n \): number of independent trials
- \( x \): number of successes
- \( p \): probability of success
- \( q \): probability of failure
Example 1: Probability of Cold
The chance of catching cold by workers in an ice factory is 25%. What is the probability that out of 5 workers, 4 or more catch cold?
Given: \( n = 5 \), \( p = 0.25 \), \( q = 0.75 \)
\[
P(X = x) = \binom{5}{x}(0.25)^x(0.75)^{5 – x}
\]
\[
P(X \geq 4) = P(X = 4) + P(X = 5)
\]
\[
P(X = 4) = \binom{5}{4}(0.25)^4(0.75)^1 = 5(0.003)(0.75) = 0.01465
\]
\[
P(X = 5) = \binom{5}{5}(0.25)^5 = 0.000977
\]
\[
P(X \geq 4) = \boxed{0.015627}
\]
Example 2: Sum of Two Binomial Variables
Let \( X \sim B(4, 0.7) \), \( Y \sim B(3, 0.7) \). Find \( P(X + Y \leq 1) \).
Since \( X \) and \( Y \) are independent:
\[
X + Y \sim B(7, 0.7)
\]
\[
P(X + Y \leq 1) = P(Z = 0) + P(Z = 1)
\]
\[
P(Z = 0) = (0.3)^7 = 0.0002187,\quad P(Z = 1) = 7(0.7)(0.3)^6 = 0.0035721
\]
\[
P(X + Y \leq 1) = \boxed{0.0037908}
\]
Example 3: Target Practice
A man hits a target with probability \( \frac{1}{4} \). What is the probability that out of 5 shots, he hits the target at least twice?
\[
P(X \geq 2) = 1 – (P(X = 0) + P(X = 1))
\]
\[
P(X = 0) = \left(\frac{3}{4}\right)^5 = \frac{243}{1024},\quad
P(X = 1) = 5 \cdot \frac{1}{4} \cdot \left(\frac{3}{4}\right)^4 = \frac{405}{1024}
\]
\[
P(X \geq 2) = 1 – \frac{648}{1024} = \frac{376}{1024} = \boxed{\frac{47}{128}}
\]
Example 4: Survival Probability
A policeman fires 6 bullets. Probability of a kill per bullet is 0.6. What is the probability that the dacoit survives?
\[
P(X = 0) = (0.4)^6 = \boxed{0.0041}
\]
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