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