Respuesta :
Answer:
If n = 1000000, then
[tex]P(h E [495000, 505000]) = \int\limits^{50500}_{495000} {\frac{2}{\sqrt{2000000\pi}} e^{\frac{-(k-500000)^2}{500000} }} \, dk[/tex]
If n = 10400, then
[tex]P(h E [495000, 505000]) = \int\limits^{50500}_{495000} {\frac{2}{\sqrt{2000000\pi}} e^{\frac{-(k-500000)^2}{500000} }} \, dk[/tex]
If N = 102, then
[tex]P(h < 40 or h > 60) = 1-P(40<h<60) = 1-\int\limits^{60}_{40} {\frac{2}{\sqrt{204\pi }} e^{\frac{-(k-51)^2}{51} }}\, dk[/tex]
Step-by-step explanation:
Since the coin is fair, then the probability that a filp is heads is 1/2. Given N tries, the amount of heads can be approximated with a Normal distribution with mean μ = N *1/2 = N/2 and standard deviation σ = √(N*1/2 * 1/2) = √N/ 2
The density function of that random variable is given by de following formula
[tex]f_X(k) = \frac{1}{\sqrt{2\pi} * \sigma} e^{\frac{-(k-\mu)^2}{2\sigma^2} } = \frac{2}{\sqrt{2\pi N}} e^{\frac{-2(k-N/2)^2}{N} }[/tex]
If n = 1000000, then
[tex]P(h E [495000, 505000]) = \int\limits^{50500}_{495000} {\frac{2}{\sqrt{2000000\pi}} e^{\frac{-(k-500000)^2}{500000} }} \, dk[/tex]
If n = 10400, then
[tex]P(h E [495000, 505000]) = \int\limits^{50500}_{495000} {\frac{2}{\sqrt{2000000\pi}} e^{\frac{-(k-500000)^2}{500000} }} \, dk[/tex]
If N = 102, then
[tex]P(h < 40 or h > 60) = 1-P(40<h<60) = 1-\int\limits^{60}_{40} {\frac{2}{\sqrt{204\pi }} e^{\frac{-(k-51)^2}{51} }}\, dk[/tex]
