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(Exercise)
The Palace is now holding a Probability Theory Examination, and you are now invited to complete the following exercises to begin your journey to promotion!
Exercise 1
Write two life examples of binomial and geometric distributions respectively.
Exercise 2
Distinguish between the following questions, whether they are Binomial or Geometric distributions, and calculate the probabilities.
Q 1:
Zhen Huan is shooting arrows, the probability of hitting the bull's-eye in archery is 75%. Calculate the probability that 4 out of 6 hits the bullseye.
Q 2:
In a game of chance, Zhen Huan flips a fair copper coin repeatedly until they get heads for the first time. Calculate the probability that she needs exactly 3 flips to achieve the first heads.
Q 3:
A factory produces porcelain bottle, and the probability of a bottle being defective is 0.05. What is the probability that the factory needs to produce 10 porcelain bottles to find the first defective one?
Q 4:
A multiple-choice test on the imperial examination has 10 questions, each with 4 possible answers and only one correct answer. If an examinee guesses randomly on each question, calculate the probability that the examinee gets exactly 3 questions correct.
Answer
Answer
Q1:
This exercise involves the binomial distribution. Using the binomial probability formula, the probability can be calculated as follows:
P(X = 4) = (6C4) * (0.75^4) * (0.25^2) ≈ 0.3115
Q 2:
This exercise involves the geometric distribution. The probability of achieving the first heads on the third flip can be calculated as follows:
P(X = 3) = (1 - 0.5)^2 * 0.5 ≈ 0.125
Q 3:
This exercise also involves the geometric distribution. The probability of finding the first defective bottle on the 10th trial can be calculated as follows:
P(X = 10) = (1 - 0.05)^9 * 0.05 ≈ 0.3774
Q 4:
This exercise involves the binomial distribution. Using the binomial probability formula, the probability of answering exactly 3 questions correctly can be calculated as follows:
P(X = 3) = (10C3) * (0.25^3) * (0.75^7) ≈ 0.2503
Feel free to use these exercises to practice and enhance your understanding of the binomial and geometric distributions.
More details about distinguishing between Binomial and Geometric distributions
Commonality:
Each experiment has only two possible outcomes, success (S) and failure (F).
The probability of success in each experiment is denoted as p, the probability of failure is 1-p.
The experiments are denoted as n, and each experiment is independent of the others.
Differences:
1. Binomial Distribution:
Describes the probability distribution of the number of successful events in a series of independent and identical experiments.
The random variable X represents the number of successful events.
The probability mass function is given by P(X=k) = C(n,k) * p^k * (1-p)^(n-k), where C(n,k) represents the binomial coefficient.
2. Geometric Distribution:
Describes the probability distribution of the number of trials needed until the first successful event occurs in a series of independent and identical experiments.
The random variable X represents the number of trials needed for the first successful event.
The probability mass function is given by P(X=k) = (1-p)^(k-1) * p, where k represents the number of trials until the first success.
Therefore, the key to differentiating between the binomial distribution and the geometric distribution is to focus on the number of occurrences of the event. If you are interested in the number of successful events within a fixed number of trials, you would use the binomial distribution. If you are interested in the number of trials needed for the first successful event, you would use the geometric distribution.
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