Answer:
The probability that the difference between the number of heads and number of tails is at most 100 is P=0.84375.
Step-by-step explanation:
Let X denote the amount of heads after 10,000 coin flips.
X will have a binomial distribution with p=0.5 and n=10,000.
When the difference between the number of heads and the number of tails is 100, thats when X=5,050: 5,050 heads and 4,950 tails.
We need to calculate the probability P(X≤5,050).
As the binomial distribution with such a large n is impractical, we use the approximation to normal
[tex]B(n,p)\approx N(np,npq)[/tex]
[tex]\mu=np=10,000*0.5=5,000\\\\\sigma=\sqrt{npq}=\sqrt{10,000*0.5*0.5}=50[/tex]
We use the continuity correction factor to make the approximation from the discrete variable to the continous variable.
[tex]P(X\leq 5050)=P(X<5,050.5)[/tex]
The z-value is
[tex]z=\frac{X-\mu}{\sigma}=\frac{5050.5-5000}{50}=\frac{50.5}{50}= 1.01[/tex]
[tex]P(X\leq 5050)=P(X<5,050.5)=P(z<1.01)=0.84375[/tex]
