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