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MHT CET202615 April 2026Morning ShiftMathematicsProbabilityActual

The centers for disease control have determined that when a person is given a vaccine, the probability that the person will develop immunity to a virus is 0.8 . If 8 people are given this vaccine, then the probability that exactly 4 will develop immunity is...

Options

  1. A(0.2)(0.8)^4
  2. B(0.112)(0.8)^4
  3. C(0.112)(0.8)^6
  4. D(0.2)^4(0.8)^4

Correct answer

B. (0.112)(0.8)^4

Step-by-step solution

Let p be the probability of developing immunity, p = 0.8 . Let q be the probability of not developing immunity, q = 1 - p = 0.2 . The number of trials is n = 8 . Using the binomial distribution formula, the probability of exactly r successes is given by P(X = r) = ^ n C_ r p^r q^ n-r . For r = 4 successes: P(X = 4) = ⁸C₄ (0.8)^4 (0.2)^4 Calculating ⁸C₄ : ⁸C₄ = 8! 4!(8-4)! = 8 7 6 5 4 3 2 1 = 70 Substituting this back into the probability expression: P(X = 4) = 70 (0.8)^4 (0.2)^4 Since (0.2)^4 = 0.0016 : 70 (0.2)^4

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