Suppose a certain program runs successfully with probability 0.80 when executed. Let $X$ be the number of executions required for a successful run.

a. Identify distribution to model $X$.

b. What is the probability that at least two attempts are required?

c. What is the probability that at most two attempts are required?

#### Solution

Let $X$ denote the number of executions required for a successful run. The random variable take the values $X=1,2,3,\cdots$.

The probability of successful run of a program when executed is $p=0.80$.

(a) The probability distribution of the random variable $X$ is a geometric distribution.

The probability mass function of random variable $X$ is

` $$ \begin{aligned} P(X=x) =\left\{ \begin{array}{ll} (0.8)*(1-0.8)^{x-1}, & \hbox{$x=1,2,\ldots$} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{aligned} $$ `

(b) The probability that at least two attempts are required is

` $$ \begin{aligned} P(X\geq 2) &= 1- P(X<2)\\ &= P(X=1)\\ &=(0.8)*(1-0.8)^{1-1}\\ &= 0.8 \end{aligned} $$ `

(c) The probability that at most two attempts are required is

` $$ \begin{aligned} P(X\leq 2) &= P(X=1) + P(X=2)\\ &=0.8*(1-0.8)^{1-1}+0.8*(1-0.8)^{2-1}\\ &= 0.8 + 0.16\\ &= 0.96 \end{aligned} $$ `

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators