Question

Question: What is the expected number of heads that come up when

a fair coin is flipped 10 times?

Short answer

Answer

The expected number of heads that come up when a fair coin is flipped 10 times is 5.

Step-by-step solution

Given Information  

A fair coin

Definition of Discrete Probability distribution  

Probability distribution Types: Discrete Probability distribution and Continuous probability distribution. They each define Discrete and Continuous random variables respectively.

Discrete Probability distribution gives the probability that a discrete random variable will have a specified value. Such a distribution will represent data that has a finite countable number of outcomes. The two conditions discrete probability distributions must satisfy are:

• \({\bf{0}}{\rm{ }} \le {\rm{ }}{\bf{P}}\left( {{\bf{X}}{\rm{ }} = {\rm{ }}{\bf{x}}} \right){\rm{ }} \le {\rm{ }}{\bf{1}}\). This implies that the probability of a discrete random variable,\({\bf{X}}\), taking on an exact value,\({\bf{x}}\), lies between\({\bf{0}}\)and\({\bf{1}}\)

• \(\sum {\bf{P}}\left( {{\bf{X}}{\rm{ }} = {\rm{ }}{\bf{x}}} \right){\rm{ }} = {\bf{1}}\). The sum of all probabilities must be equal to\({\bf{1}}\)

Some commonly used Discrete probability distributions are Geometric distributions, binomial distributions and Bernoulli distributions

Calculation of the expected number of heads that come up when a fair coin is flipped 10 times  

We are interested in the expected number of heads among 5 trials.

\(n = 10\)

The coin is fair and then have 1 chance in 2 to flip heads.

\(p = \frac{1}{2}\)

The expected number of success (heads) among a fixed \(n\)of mutually independent Bernoulli trials is\(np\), with\(p\)the probability of success

\(E\left( X \right) = np\)

\(\begin{array}{l} = 10 \times \frac{1}{2}\\ = 5\end{array}\)

Therefore, on flipping a fair coin 10 times, the expected number of heads that come up is 5