**There are a total n=8 number of trials with the probability p=0.20 of being successful or correct. Find the probability for the number of correct answers.**

**Find the probability that the total number of correct answers is less than 3.**In this question, we have to find two things, first the

**probability**of how much

**correct answers**and secondly the

**probability**of the

**correct answers**being

**less than**3. The basic concept behind this question is a sound knowledge of

**Statistics**and the concept of

**Probability**and precisely

**binomial probability**. To solve this question we will apply the concept of

**binomial probability.**We know that

**binomial probability**is represented as follows: \[ P(x)\ = {_\ ^n}C_x \ (p )^x ( 1 – p )^{ n – x } \]

## Expert Answer

Given in the question statement, we have: \[ n = 8 \] \[ p = 0.20 \] We know that**binomial probability**is expressed as follows: \[ P(x)\ = {_\ ^n}C_x \ (p )^x ( 1 – p )^{ n – x } \] Let us suppose $ P ( x ) = 0$, putting $ n= 8$ and $ p= 0.20$ we will have the

**binomial probability**for this as follows: \[ P (0)\ = {_\ ^8}C_0 \ (0.20 )^0 ( 1 – 0.20 )^{ 8 -0 } \] \[ P (0)\ = 0.168 \] Let us suppose $ P ( x ) = 1$, putting $ n= 8$ and $ p= 0.20$, we will have the

**binomial probability**for this as follows: \[ P (1)\ = {_\ ^8}C_1 \ (0.20 )^1 ( 1 – 0.20 )^{ 8 -1 } \] \[ P (1)\ = 0.335 \] Let us suppose $ P ( x ) = 2$, putting $ n= 8$ and $ p= 0.20$, we will have the

**binomial probability**for this as follows: \[ P (2)\ = {_\ ^8}C_2 \ (0.20 )^2 ( 1 – 0.20 )^{ 8 -2 } \] \[ P (2)\ = 0.294 \] Now to find if the

**probability**that the total number of

**correct answers**$x $ is

**less than**$3$, we write the

**binomial probability**equation as follows: \[ P ( less \space than \space 3) = P (0) + P (1) + P (2) \] Here, we will put the equations and values of $P (0)$, $P (1)$ and $P (2)$: \[P(0)\ =0.168\] \[P(1)\ =0.335\] \[P(2)\ =0.294\] \[P(less \space than \space 3)=0.168 + 0.335 + 0.294\] \[P(less \space than \space 3)=0.797 \]

## Numerical Results

The**probability**that the total number of

**correct answers**$x$ is less than $3$ is

**0.797**. \[ P ( less \space than \space 3) = 0.797\]

## Example

There are a total $n=8$**number of trials**with the probability $p=0.20$ of being correct. Find the probability that the total number of

**correct answers**is less than $2$. \[ P ( less \space than \space 2) = P (0) + P (1)\] \[ P ( less \space than \space 2) = 0.168 + 0.335\] \[ P ( less \space than \space 2) = 0.503\]

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