Question

Find the indicated probability. Round to three decimal places. Show work
In a certain college, 33% of the physics majors belong to ethnic minorities. If 10 students are selected at random from the physics majors, that is theprobability that no more than 6 belong to an ethnic minority?

Answer 1

This can be done as a binomial probability as

(1) The trials are independent

(2) There are only 2 mutually exclusive outcomes and

(3) The probabilities are same for every trial.

Please note ⁿC_x = n! / [(n-x)!*x!]

Binomial Probability = ⁿC_x * (p)^x * (q)^n-x, where n = number of trials and x is the number of successes.

Also sum of probabilities from 0 till n = 1, i.eP(0) + P(1) + P(2) +.......+P(n) = 1

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Here n = 10, p = 33% = 0.33, q = 1 – p = 0.67

To Find P(no more than 6) = P(X 6) = P(0) + P(1) + .....+P(6) .

Since P(0) + P(1) + ......+ P(10) = 1,

Therefore P(0) + P(1) + .....+P(6) = 1 - [P(7) + P(8) + P(9) + P(10)

P(X = 7) = ¹⁰C₇ * (0.33)⁷ * (0.67)^{10-7 = 3} = 0.01538

P(X = 8) = ¹⁰C₈ * (0.33)⁸ * (0.67)^{10-8 = 2} = 0.00284

P(X = 9) = ¹⁰C₉ * (0.33)⁹ * (0.67)^{10-9 = 1} = 0.00031

P(X = 10) = ¹⁰C₁₀ * (0.33)¹⁰ * (0.67)^{10-10 = 0} = 0.00002

Therefore P(X 6) = 1 - (0.01532 + 0.00284 + 0.00031 + 0.00002 ) = 1 - 0.01855 = 0.98145

Therefore P(X 6) = 0.9815 (rounding to 4 decimal places)