Asked by Julius
Cartons of milk from a particular supermarket are advertised as containing 1 litre of milk, but in fact the volume of the milk in a carton is normally distributed with mean 1012 and standard deviation 5ml.
(i) Find the probability that a randomly chosen carton contains more than 1010ml.
(ii) Find the probability that exactly 3 cartons in a sample of 10 cartons contain more than 1012ml.
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Answered by
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Let X ~ N(μ = 1012 ml, σ = 5 ml).
(i) P(X > 1010) = 1 − Φ((1010 − 1012)/5) = 1 − Φ(−0.4) = Φ(0.4) ≈ 0.6554.
(ii) For each carton p = P(X > 1012) = 1 − Φ(0) = 0.5. For n = 10, k = 3:
P(exactly 3) = C(10,3) p^3 (1 − p)^{7} = 120 · 0.5^{10} = 120/1024 ≈ 0.1172.
(i) P(X > 1010) = 1 − Φ((1010 − 1012)/5) = 1 − Φ(−0.4) = Φ(0.4) ≈ 0.6554.
(ii) For each carton p = P(X > 1012) = 1 − Φ(0) = 0.5. For n = 10, k = 3:
P(exactly 3) = C(10,3) p^3 (1 − p)^{7} = 120 · 0.5^{10} = 120/1024 ≈ 0.1172.
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