Question

Tossing a Coin. If we repeatedly toss a balanced coin, then, in the long run, it will come up heads about half the time. But what is the probability that such a coin will come up heads exactly half the time in 10 tosses?

Short answer

The probability that heads come up exactly half the time in 10 tosses is 0.246.

Step-by-step solution

Step 1. Given information.

The statement given in the question is:

If we repeatedly toss a balanced coin, then, in the long run, it will come up heads about half the time.

Step 2. Find the probability.

The total number of throws is:

$n=10$

The probability of receiving a head while tossing a balanced coin is:

$\frac{1}{2}$

Therefore, the likelihood of success is$\frac{1}{2}$.

Step 3.  Find the probability.

The formula for binomial probability is:

$P\left(X=x\right)=\left(\begin{array}{c}n\\ x\end{array}\right)p^x{\left(1-p\right)}^{n-x}$

Let Xbe the total number of successful Bernoulli trials with probability in nBernoulli trials.

$$\begin{gathered} P\left(X=5\right)=\left(\begin{array}{c}10\\ 5\end{array}\right){\left(\frac{1}{2}\right)}^5{\left(1-\frac{1}{2}\right)}^{10-5} \\ =\frac{10!}{5!\left(10-5\right)!}{\left(\frac{1}{2}\right)}^5{\left(\frac{1}{2}\right)}^5 \\ =252\left(0.03125\right)\left(0.03125\right) \\ =0.246 \end{gathered}$$