Question

Suppose that a fair coin is tossed 10 times independently.

Determine the p.f. of the number of heads that will be obtained.

Short answer

The required probability function is,

\(f\left( x \right) = \left\{ \begin{array}{l}\left( \begin{array}{l}10\\x\end{array} \right){\left( {\frac{1}{2}} \right)^{10}};\;\;x = 0,1,...,10\\0\;\;\;\;\;\;\;\;\;\;\;\;\;;\;\;\;otherwise\end{array} \right.\)

Step-by-step solution

Given information

A fair coin is tossed 10 times independently.

Determine the probability function

The number of times the coin is tossed is, \(n = 10\).

It is known that the probability of getting a head is, \(p = \frac{1}{2}\).

Let X be the random variable representing the number of heads.

In the given scenario, the random variable X will follow the binomial distribution as the trials are independent and there are only two possible outcomes (a head or a tail).

The probability function of a binomial distribution is given as,

\(f\left( x \right) = \left\{ \begin{array}{l}\left( \begin{array}{l}n\\x\end{array} \right){p^x}{\left( {1 - p} \right)^{n - x}}\;\;for\;x = 0,1,...,n,\\0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise\end{array} \right.\)

where there are n trials with p probability of success in each trial..

For the provided scenario, the probability function of X is,

\(f\left( x \right) = \left\{ \begin{array}{l}\left( \begin{array}{l}10\\x\end{array} \right){\left( {\frac{1}{2}} \right)^{10}};\;\;\;x = 0,1,...,10\\0\;\;\;\;\;\;\;\;\;\;\;\;\;;\;\;\;\;otherwise\end{array} \right.\)