Answer to Question #253057 in Statistics and Probability for Malika

Question

1)                 Plastic is now surpassing aluminum as the packaging material of choice for soft drinks. According to a recent article, 80% of all new vending business is going to plastic. If a random sample of 10 new vending businesses is selected, what are P(X>2), P(X<5), and P(X<9)?

Answer 1

$p=0.8 \\ q = 1-p = 0.2 \\ n = 10 \\ P(X=x) = \binom{n}{x}p^xq^{n-x} \\ P(X>2) = 1- P(X≤2) = 1 -[P(X=0) + P(X=1) + P(X=2)] \\ P(X=0) = \frac{10!}{0!(10-0)!} \times 0.8^0 \times 0.2^{10-0} = 1.02 \times 10^{-7}\\ P(X=1) = \frac{10!}{1!(10-1)!} \times 0.8^1 \times 0.2^{10-1} = 40.96 \times 10^{-7} \\ P(X=2) = \frac{10!}{2!(10-2)!} \times 0.8^2 \times 0.2^{10-2} = 437.28 \times 10^{-7} \\ P(X>2) = 1- 479.26 \times 10^{-7} = 0.99995$

$P(X<5) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) \\ P(X=3) = \frac{10!}{3!(10-3)!} \times 0.8^3 \times 0.2^{10-3} = 0.000688 \\ P(X=4) = \frac{10!}{4!(10-4)!} \times 0.8^4 \times 0.2^{10-4} = 0.004128 \\ P(X<5) = 0.004864$

$P(X<9) = 1 -[P(X=9) + P(X=10)] \\ P(X=9) = \frac{10!}{9!(10-9)!} \times 0.8^9 \times 0.2^{10-9} = 0.268435 \\ P(X=10) = \frac{10!}{10!(10-10)!} \times 0.8^10 \times 0.2^{10-10} = 0.107374 \\ P(X<9) = 1 -0.375809 =0.624191$