Respuesta :
ANSWER:
The probability for 13 or more of the 15 trees they just planted will survive is 0.00110702414
SOLUTION:
Given,
The total number of plants is, n=15
The chance of survival of trees is, p=45%.
Probability of getting success p(success) = [tex]\frac{n o \text { of favourable outcomes }}{\text {total possible outcomes}}[/tex]
P(success) = [tex]\frac{45}{100}[/tex]
P(success) = 0.45
Binomial distribution formula is given as
[tex]\mathrm{P}(\mathrm{X}=\mathrm{x})=\mathrm{n}_{\mathrm{C} \mathrm{x}}(\mathrm{p})^{\mathrm{x}} \cdot(1-\mathrm{p})^{\mathrm{n}-\mathrm{x}}[/tex]
In our case, x is greater than or equal to 13, i.e. x [tex]\geq[/tex] 13
The probability for 13 or more of the 15 trees they just planted will survive is given by
[tex]\mathrm{P}(\mathrm{X} \geq 13)=\mathrm{P}(\mathrm{X}=13)+\mathrm{P}(\mathrm{X}=14)+\mathrm{P}(\mathrm{X}=15)[/tex]
[tex]\mathrm{P}(\mathrm{X} \geq 13)=\left(15 \mathrm{C}_{13} \times(0.45)^{13} \times(1-0.45)^{15-13}\right.[/tex] + [tex]\left(^{15} \mathrm{C}_{14} \times(0.45)^{14} \times(1-0.45)^{15-14}\right.[/tex] + [tex]\left(15 \mathrm{C}_{15} \times(0.45)^{15} \times(1-0.45)^{15-15}\right.[/tex]
on simplification we get
[tex]=(15 \times 7) \times(0.45)^{13} \times(0.55)^{2}+15 \times(0.45)^{14} \times(0.55)^{1}+1 \times(0.45)^{15} \times 1[/tex]
[tex]\begin{array}{c}{=(105 \times 0.00003102863 \times 0.3025)+(15 \times 0.00001396288 \times 0.55)+} \\ {0.00000628329}\end{array}[/tex]
=0.00098554703 + 0.0001151938 + 0.00000628329
= 0.00110702414
Hence, the probability for 13 or more of the 15 trees they just planted will survive is 0.00110702414
