Binomial and Poisson Distributions The Binomial distribution answers: "In $n$ independent trials (each with probability $p$ of success), what is the probability of exactly $x$ successes?" Requirements: fixed number of trials, only two outcomes per trial (success/failure), same $p$ each time, and trials are independent. The Poisson distribution models how many times an event occurs in a fixed interval when the average rate is $\lambda$. Use it when events happen randomly over time or space (calls per hour, defects per meter of pipe, accidents per month). Mean and variance of Binomial: $\mu = np$, $\sigma^2 = np(1-p)$. For Poisson: $\mu = \sigma^2 = \lambda$.
$$\text{Binomial: }P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}, \quad \mu=np$$
$$\text{Poisson: }P(X=x)=\frac{e^{-\lambda}\lambda^x}{x!}, \quad \mu=\lambda$$
Binomial Defective Items A batch has 5% defective items. If 10 items are selected, find the probability exactly 2 are defective.
Show Solution $$P(X=2)=C(10,2)(0.05)^2(0.95)^8=0.0746$$
Final answer: 0.0746.
At Least One Success A machine has 0.90 probability of passing inspection. If 4 machines are inspected, find the probability at least one fails.
Show Solution At least one fails is the complement of all pass.
$$P=1-(0.90)^4=0.3439$$
Final answer: 0.3439.
Poisson Arrivals Calls arrive at an average of 3 per hour. Find the probability of exactly 5 calls in one hour.
Show Solution $$P(X=5)=\frac{e^{-3}3^5}{5!}=0.1008$$
Final answer: 0.1008.
Binomial Mean and Variance A fair coin is flipped 20 times. Find the expected number of heads and the standard deviation.
Show Solution $n=20$, $p=0.5$, $q=0.5$.
$$\mu=np=20(0.5)=10$$
$$\sigma^2=np(1-p)=20(0.5)(0.5)=5, \quad \sigma=\sqrt{5}\approx2.24$$
Final answer: Expected heads = 10, σ ≈ 2.24.
Binomial: P(X ≥ 3) The probability a concrete cylinder passes a strength test is 0.8. If 5 cylinders are tested, find P(X ≥ 3).
Show Solution Find $P(X\geq3)=P(X=3)+P(X=4)+P(X=5)$, or use the complement: $1-P(X\leq2)$.
$$P(X=3)=C(5,3)(0.8)^3(0.2)^2=10(0.512)(0.04)=0.2048$$
$$P(X=4)=C(5,4)(0.8)^4(0.2)^1=5(0.4096)(0.2)=0.4096$$
$$P(X=5)=(0.8)^5=0.3277$$
$$P(X\geq3)=0.2048+0.4096+0.3277=0.9421$$
Final answer: 0.9421.
Poisson: Probability of Zero Events Accidents occur at a construction site at a rate of 2 per month. What is the probability of no accidents in a given month?
Show Solution $\lambda=2$, $x=0$. Recall $e^{-2}\approx0.1353$.
$$P(X=0)=\frac{e^{-2}(2)^0}{0!}=e^{-2}=0.1353$$
Final answer: 0.1353 or about 13.5% chance of a zero-accident month.
Poisson: Changing the Interval Defects occur at a rate of 4 per 100 meters of cable. Find the probability of exactly 2 defects in 50 meters.
Show Solution Rescale the rate: for 50 m, $\lambda=4\times(50/100)=2$.
$$P(X=2)=\frac{e^{-2}(2)^2}{2!}=\frac{0.1353(4)}{2}=0.2707$$
Final answer: 0.2707.
Quality Control with Binomial A lot has 10% defective items. A sample of 8 is inspected. Find P(exactly 1 defective) and P(at most 1 defective).
Show Solution $n=8$, $p=0.10$, $q=0.90$.
$$P(X=0)=C(8,0)(0.1)^0(0.9)^8=0.4305$$
$$P(X=1)=C(8,1)(0.1)^1(0.9)^7=8(0.1)(0.4783)=0.3826$$
$$P(X\leq1)=0.4305+0.3826=0.8131$$
Final answer: P(exactly 1) = 0.3826; P(at most 1) = 0.8131.
