Question

Question: What is the expected number times a 6 appears when a fair die is rolled 10 times?

Short answer

Answer

The expected number times a 6 appears when a fair die is rolled 10 times is 1.667

Step-by-step solution

Given Information  

A fair die

Definition of Integral and Integration  

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 times a 6 appears when a fair die is rolled 10 times

The expected number of successes for\(n\)Bernoulli trials is\(np\).In the present problem we have,

\(n = 10,p = \frac{1}{6}\)

Therefore, the expected number of successes ((i.e.) appearance of a 6) is

\(E(X) = np\)

\(\begin{array}{l} = 10 \times \frac{1}{6}\\ = 1.667\end{array}\)

Therefore, on rolling a fair die 10 times, the expected number times a 6 appears is 1.667