Question: If a researcher studies 4 random integers between 0 and 10 inclusive, what is the probability that all are even?
["Understanding the Probability: When 4 Random Integers from 0 to 10 Are All Even", "When studying random sampling, one common question arises: If a researcher selects 4 random integers between 0 and 10 inclusive, what is the probability that all four numbers are even? This question combines basic principles of probability and number theory, offering clear insights into random integer selection.", "### Breakdown of the Problem", "The integers range from 0 to 10, inclusive — a total of 11 possible values:\n0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.", "Among these, the even integers are:\n0, 2, 4, 6, 8, 10 — a total of 6 even numbers.", "The odd numbers are:\n1, 3, 5, 7, 9 — which make up the remaining 5 integers.", "### Probability Calculation", "Assuming each integer is selected independently and uniformly at random from the 11 values (with replacement), the probability of selecting an even integer is:", "[\nP(\ ext{even}) = \frac{\ ext{Number of even integers}}{\ ext{Total integers}} = \frac{6}{11}\n]", "Since the selections are independent, the probability that all four chosen integers are even is:", "[\nP(\ ext{all even}) = \left( \frac{6}{11} \right)^4\n]", "Calculating this:", "[\n\left( \frac{6}{11} \right)^4 = \frac{6^4}{11^4} = \frac{1296}{14641} \approx 0.0884\n]", "So, the probability is approximately 8.84%.", "### Why Integer Selection Matters in Research", "Understanding such probabilities is crucial in research fields involving simulations, statistical modeling, or random sampling. It helps researchers assess randomness, validate models, and interpret data distributions correctly.", "### Summary", "- Total numbers (0–10): 11\n- Even numbers: 6\n- Probability of one even number: ( \frac{6}{11} )\n- Probability all four are even: ( \left( \frac{6}{11} \right)^4 \approx 8.84% )", "This problem exemplifies how combinatorics and basic probability converge in random integer generation — key knowledge for researchers and data scientists alike.", "---", "Keywords: probability of even numbers, 4 random integers 0 to 10, sample probability calculation, discrete uniform distribution, math research probability, random sampling theory"]
