
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 $2$. \[ P ( less \space than \space 2) = P (0) + P (1)\] \[ P ( less \space than \space 2) = 0.168 + 0.335\] \[ P ( less \space than \space 2) = 0.503\]Previous Question < > Next Question
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