A student takes 10-questions, true/false exam and guesses on each question. a. Find the...

Question:

A student takes 10-questions, true/false exam and guesses on each question.

a. Find the probability of passing If the lowest passing grade is 6 correct of 10.

b. Based on your answer, would it be a good idea not to study and to depend on guessing?

Binomial Distribution:

It is a discrete probability distribution whose probability mass function is given by:

{eq}{\rm{P}}\left( {X = x} \right) = {}^n{C_x}{\left( p \right)^x}{\left( {1 - p} \right)^{n - x}} {/eq}

For x =0, 1, 2,...., n

{eq}{\rm{where}}\;{}^n{C_x} = \;\dfrac{{n!}}{{x!\left( {n - x} \right)!}} {/eq}

Where n = The number of times an experiment is performed.

and p= the probability of Success.

Answer and Explanation:

Given Information:

A student takes 10-questions, true/false exam and guesses on each question. The chance that he will get correct answer by guessing is 1(True) out of 2(True and False).

Let X denotes the event of number of questions correctly answered by guessing. Then

{eq}\begin{align*} X &\sim Bin\left( {10,\dfrac{1}{2}} \right)\\ {\rm{P}}\left( {X = x} \right) &= {}^{10}{C_x}{\left( {\dfrac{1}{2}} \right)^x}{\left( {1 - \dfrac{1}{2}} \right)^{10 - x}}\\ {\rm{P}}\left( {X = x} \right) &= {}^{10}{C_x}{\left( {\dfrac{1}{2}} \right)^{10}} \end{align*} {/eq}

a) The probability of passing if the lowest passing grade is 6 correct of 10 is given by:

{eq}\begin{align*} P\left( {X \ge 6} \right) = 1 - P\left( {X < 5} \right)\\ &= 1 - \left[ {P\left( {X = 0} \right) + P\left( {X = 1} \right) + P\left( {X = 2} \right) + P\left( {X = 3} \right) + P\left( {X = 4} \right)} \right]\\ &= 1 - \left[ {{}^{10}{C_0}{{\left( {\dfrac{1}{2}} \right)}^{10}} + {}^{10}{C_1}{{\left( {\dfrac{1}{2}} \right)}^{10}} + {}^{10}{C_2}{{\left( {\dfrac{1}{2}} \right)}^{10}} + {}^{10}{C_3}{{\left( {\dfrac{1}{2}} \right)}^{10}} + {}^{10}{C_4}{{\left( {\dfrac{1}{2}} \right)}^{10}}} \right]\\ &= 1 - 0.000976563 \times \left[ {1 + 10 + 45 + 120 + 210} \right]\\ &= 1 - 0.376953318\\ &= 0.62304668 \end{align*} {/eq}

The probability of passing if the lowest passing grade is 6 correct of 10 is 0.62304668.

b) No, it will not be a good idea not to study and to depend on guessing as the probability is greater than 0.5 but less than 1, you must study for exam and must not rely on guessing as the probability is not closer to 1.


Learn more about this topic:

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What is the Binomial Theorem?

from Math 101: College Algebra

Chapter 11 / Lesson 3
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