In this topic you match a problem to a named distribution and use its stored formula. You check the conditions, set the parameters, and compute. A problem belongs here when it fits one of four setups. Counting successes in a fixed number of trials is binomial. Waiting for the first success is geometric. Counting events at a steady average rate is Poisson. A continuous measurement with a supplied table is normal. You will practice: Binomial distribution, Geometric distribution, Normal distribution and standardizing, Poisson distribution and thinning.
A router sends 4 packets across a congested link. Each packet is dropped independently with probability 0.2. What is the probability that exactly 1 of the 4 packets is dropped?
Answer:
The number of dropped packets is binomial with n = 4 and p = 0.2.
You roll a fair six-sided die repeatedly until you get the first six. What is the probability that the first six appears on exactly the third roll?
Answer:
The roll number of the first six is geometric with p = 1/6.
A gardener plants 5 tomato seeds. Each seed germinates independently with probability 0.7. What is the probability that exactly 4 of the 5 seeds germinate?
Answer:
The number of germinating seeds is binomial with n = 5 and p = 0.7.
Each scratch-off lottery ticket from a large roll is a winner independently with probability 0.25. You scratch tickets one at a time. What is the probability that your first winning ticket is exactly the 4th one you scratch?
Answer:
The trial number of the first win is geometric with p = 0.25.
An archer shoots 3 arrows at a target. Each arrow hits the bullseye independently with probability 0.9. What is the probability that all 3 arrows hit the bullseye?
Answer:
The number of bullseyes is binomial with n = 3 and p = 0.9.
A telemarketer calls households one at a time. Each call results in a sale independently with probability 0.1. What is the probability that the first sale occurs on exactly the fourth call?
Answer:
The call number of the first sale is geometric with p = 0.1.
A fair coin is tossed 4 times. The tosses are independent. What is the probability that all 4 tosses land heads?
Answer:
Each toss is heads with probability 1/2, so multiply the four probabilities.
A spinner lands on red with probability 1/3 on each spin. Spins are independent. You spin until red appears for the first time. What is the probability that the first red is on exactly the second spin?
Answer:
Geometric pattern: one failure, then the first success.
A basketball player makes each free throw independently with probability 0.6. She takes 5 free throws. What is the probability that she makes exactly 3 of them?
Answer:
The number of makes is Binomial(5, 0.6). Use the binomial pmf at k = 3.
A bakery receives custom cake orders at an average rate of 2 orders per day. The number of orders in a day follows a Poisson distribution. What is the probability that the bakery receives exactly 3 orders tomorrow?
Answer:
Use the Poisson pmf with λ = 2 at k = 3.
The heights of adult women in a city are normally distributed with mean μ = 165 cm and standard deviation σ = 5 cm. What is the probability that a randomly chosen woman is taller than 170 cm? Use Φ(1) = 0.8413.
Answer:
Standardize 170 and take the upper tail.
A wildlife camera is set up in a forest. On each night, independently, it captures a photo of a fox with probability 0.2. What is the probability that the first fox photo happens after night 3, that is, the first 3 nights all have no fox photo?
Answer:
The first success comes after night 3 exactly when the first 3 nights are all failures.
Want these problems to schedule themselves? Inkerval brings handwritten practice and spaced repetition together on iPad.
Be first in when the doors open. No spam, ever.