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lakkis [162]
3 years ago
13

21​% of adults would pay more for environmentally friendly products. You randomly select 10 adults. Find the probability that th

e number of adults who would pay more for environmentally friendly products is​ (a) exactly​ two, (b) more than​ two, and​ (c) between two and​ five, inclusive.
Mathematics
1 answer:
ElenaW [278]3 years ago
4 0

Answer: a) 0.3011

b) 0.3526

c) 0.6455

Step-by-step explanation:

Given : The proportion of adults would pay more for environmentally friendly products : p= 0.21

Sample size : n= 10

Let x be a binomial variable that denotes the number of adults would pay more for environmentally friendly products.

Using binomial distribution, P(x)=^nC_xp^x(1-p)^{n-x}

a) The probability that the number of adults who would pay more for environmentally friendly products is exactly 2 will be :-

P(x=2)=^{10}C_2(0.21)^2(1-0.21)^{10-2}\\\\=\dfrac{10!}{2!(10-2)!}(0.21)^2(0.79)^8\approx0.3011

The probability that the number of adults who would pay more for environmentally friendly products is exactly 2=0.3011

b) The probability that the number of adults who would pay more for environmentally friendly products is more than​ two will be :-

P(x>2)=1-P(x\leq2)=1-(P(x=0)+P(x=1)+P(x=2))\\\\=1-(^{10}C_0(0.21)^0(1-0.21)^{10}+^{10}C_1(0.21)^1(1-0.21)^{9}+0.3011)\\\\=1-((0.79)^{10}+(10)(0.21)(0.79)^9+0.3011)\\\\=1-(0.0946+0.2517+0.3011)\\\\=1-0.6474=0.3526

The probability that the number of adults who would pay more for environmentally friendly products is more than​ two =0.3526

c) The probability that the number of adults who would pay more for environmentally friendly products is between two and​ five, inclusive will be :-

P(2\leq x\leq5)=P(x=2)+P(x=3)+P(x=4)+P(x=5)\\\\=0.3011+^{10}C_3(0.21)^3(1-0.21)^7}+^{10}C_4(0.21)^4(1-0.21)^{6}+^{10}C_5(0.21)^5(1-0.21)^{5})\\\\=0.3011+\dfrac{10!}{3!7!}(0.21)^3(0.79)^7+\dfrac{10!}{4!6!}(0.21)^4(0.79)^6+\dfrac{10!}{5!5!}(0.21)^5(0.79)^5\\\\=0.3011+0.2134+0.0993+0.0317=0.6455

The probability that the number of adults who would pay more for environmentally friendly products is between two and​ five, inclusive =0.6455

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