A true coin is tossed 104 times.

(a) Find the probability of getting exactly 5000 heads.

(b) Find the probability of between4900and 5075 heads.

A coin is tossed 104 times.

Frequency distribution of the number of successful outcomes that can be achieved in a given number of trials, each with an equal chance of success.

(a)

The formula for binomial distribution states that.$f\left(x\right)=C(x,n)p^x{q}^{n-x}$

Substitute the values mentioned above in the formula.

$\begin{array}{c}f\left(5000\right)=C({10}^4,5\cdot {10}^3){\left(\frac{1}{2}\right)}^{{10}^4}\\ =0.008\end{array}$

The formula for corresponding normal approximation states that .$f\left(x\right)=\frac{e^{-{(x-np)}^2/2npq}}{\sqrt{2\pi npq}}$

$\begin{array}{c}n={10}^4\\ p=0.5\\ q=0.5\\ x=5000\end{array}$

Substitute the values mentioned above in the formula.

$\begin{array}{c}f\left(5000\right)=\frac{1}{\sqrt{2\pi \times {10}^4\times 0.25}}\\ f\left(5000\right)=0.0079\end{array}$

(b)

The probability by cumulative function is given below.

$P(4900\le x\le 5075)=0.9123$

Theprobability by normal distribution function is given below.

$P(4900\le x\le 5075)=0.9014$

The value required for part a is given below.

$\begin{array}{c}f\left(5000\right)=0.008\\ f\left(5000\right)=0.0079\end{array}$

The value required for part b is given below.

$\begin{array}{c}P(4900\le x\le 5075)=0.9123\\ P(4900\le x\le 5075)=0.9014\end{array}$