Question

A quiz consists of 20 multiple-choice questions, each with 4 possible answers. For someone who makes random guesses for all of the answers, find the probability of passing if the minimum passing grade is 50 %

Answer 1

Answer:

The probability of passing = 0.0127

Step-by-step explanation:

To find the probability of passing if the minimum passing grade is 50%, we can use the Binomial Distribution.

Given:

• number of trials ($n$) = 20
• probability of success ($p$) = $\frac{1}{4}$ = 0.25
• probability of failure ($q$) = $1-p$ = 0.75

Let:

X = passing, which means total of success trial ≥ 10, then P(X) = P(x ≥ 10). However without computer or graphic calculator, it would be too tedious if we use the Binomial Distribution. Then, we check these following rules if we can use the Normal Distribution for Binomial:

• $np > 5$
• $nq > 5$

In this question:

$np=20\times0.25$

    $=5$

$nq=20\times0.75$

    $=15$

Then we can use the Normal Distribution for Binomial:

• mean ($\mu$) = $np$

                       = $20\times0.25$

                       = $5$

• standard deviation ($\sigma$) = $\sqrt{npq}$

                                             = $\sqrt{20\times0.25\times0.75}$

                                             = $\sqrt{3.75}$

Since Normal Distribution is a continuous distribution, whereas Binomial Distribution is a discrete distribution, then we have to do the Continuity Correction, where:

$\boxed{P(x\geq a)\ has\ to\ convert\ to\ P(x\geq (a-0.5))}$

Hence:

$P(x\geq 10)\ has\ to\ convert\ to\ P(x\geq 9.5)$

First, we have to find the Z-score:

$\boxed{Z=\frac{x-\mu}{\sigma} }$

$\displaystyle Z=\frac{9.5-5}{\sqrt{3.75} }$

$Z=2.324$

By using the Normal distribution table, we can find that:

$P(Z\geq 2.234)=0.0127$