Chapter 6, Problem 15TP

Solution

To State:

The probability rounded to the nearest thousandth of tossing a coin $50$ times and getting exactly $15$ heads.

  $0.002$

Given:

The event that exactly $15$ heads occur out of total $50$ tosses is considered.

Concepts Used:

The probability to get a head in one toss of a coin is $\frac{1}{2}$ .

The result of each toss is independent of the result of the other tosses.

In a binomial experiment with $n$ independent trials each having probability of success $p$ . The probability of $r$ successes is given by ${}_n{C}_r{p}^r{\left(1-p\right)}^{n-r}$ .

The definition ${}_n{C}_r=\frac{n!}{r!\left(n-r\right)!}$ .

Calculations:

Calculate the probability rounded to the nearest thousandth of tossing a coin $50$ times and getting exactly $15$ heads. The experiment is equivalent to a binomial experiment with $50$ independent trials each having probability of success $\frac{1}{2}$ .

  $\begin{array}{c}P\left(\text{exactly 15 heads}\right){=}_{50}{C}_{15}{\left(\frac{1}{2}\right)}^{15}{\left(1-\frac{1}{2}\right)}^{50-15}\\ =\frac{50!}{15!\left(50-15\right)!}{\left(\frac{1}{2}\right)}^{50}\\ \approx \text{0}\text{.002}\end{array}$

Conclusion:

  $0.002$ is the probability rounded to the nearest thousandth of tossing a coin $50$ times and getting exactly $15$ heads.