Probability of Ethnic Minorities in a Sample of Physics Majors

Probability of Ethnic Minority Among Physics Majors

Problem Overview

  • A college has 33% of physics majors belonging to ethnic minorities.
  • A scenario is set where 10 students are selected at random from the physics majors.
  • Objective: Calculate the probability that no more than 6 of selected students belong to an ethnic minority.

Given Data

  • Proportion of physics majors who belong to ethnic minorities:
    • p=0.33p = 0.33
  • Proportion of physics majors who do not belong to ethnic minorities:
    • q=1−p=0.67q = 1 - p = 0.67
  • Number of students selected:
    • n=10n = 10

Probability Calculation

  • The situation follows a binomial distribution because:
    • Each selection is independent.
    • Each student has the same probability of belonging to an ethnic minority.
Formula for Binomial Distribution
  • The probability of having exactly k successes (students belonging to ethnic minorities) in n trials (total students selected) is given by the binomial probability formula:
    • P(X=k)=(nk)pkqn−kP(X = k) = {n \choose k} p^k q^{n-k}
    • Where:
    • (nk){n \choose k} = binomial coefficient
    • pp = probability of success (student belongs to a minority)
    • qq = probability of failure (student does not belong to a minority)
Required Calculation
  • We seek the cumulative probability of having no more than 6 students from ethnic minority, i.e., calculate:
    • P(X≤6)=P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)+P(X=6)P(X \leq 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)
Individual Probabilities
  • To compute each of these probabilities, utilize the formula:
    • For example, for $k = 0$:
    • P(X=0)=(100)(0.33)0(0.67)10=1⋅1⋅(0.67)10P(X = 0) = {10 \choose 0} (0.33)^0 (0.67)^{10} = 1 \cdot 1 \cdot (0.67)^{10}
    • Similarly compute for $k = 1$, $k = 2$, …, $k = 6.
Continuing Calculations
  • These need to be calculated to find their respective values.

Final Calculation

  • After calculating the probabilities for k = 0 to k = 6 and summing them up, round the final answer to three decimal places.

Possible Options

  • Overall options provided for the final answer include:
    • A. 0.981
    • B. 0.913
    • C. 0.055
    • D. 0.985

Conclusion

  • The answer should be derived based on cumulative probabilities and compared against the provided options to select the most accurate answer.