Answer :
Answer:
5% is the smallest probability
Step-by-step explanation:
5% = 0.05
2000 x 0.05 = 100 so the smallest chance is 5% but it could be more if you hit more bullseye
This is an example of a Binomial distribution, in which the chance of striking the dart here on the disc is:
[tex]=\frac{\text{area of the small disk} }{ \text{Area of the bigger disk}}[/tex]
[tex]= \frac{\pi\times R_1^2}{\pi \times R_2^2} \\\\ = \frac{1^2}{5^2} \\\\ = \frac{1}{25} \\\\ = 0.04[/tex]
Since this is a huge figure, one can use the central limit theory to approximate a supplied binomial distribution to a normally distributed one. As just a result, you have:
[tex]X \sim Bin(2000,0.04) n=2000 \ and\ p = 0.04[/tex]
Since this is [tex]np = 2000 \times 0.04 = 80[/tex] a huge figure, one can use the central limit theory to approximate a supplied binomial distribution to a normally distributed one. As just a result, you have:
[tex]X \sim N(np,np(1-p)) \\\\X \sim N(2000\times 0.04,2000 \times 0.04 \times (1-0.04)) \\\\X \sim N(80,76.8)[/tex]
We must now calculate the likelihood of hitting the bullseye at least 100 times:
[tex]P(X\geq 100)[/tex]
When we convert this to a conventional normal variable, we get:
[tex]P(Z \geq \frac{100-80}{\sqrt{76.8}})\\P(Z \geq 2.2822)[/tex]
When we use the conventional normal table to calculate this probability, we get:
[tex]P(Z \geq 2.2822) = 0.0112[/tex]
Therefore the required probability is "0.0112".
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