Mathematics: 98 points in total! 1 question! no calculator! show your work! a student randomly guesses on 10 true or false questions. use the binomial model to determine the probability that the student gets 5 out of the 10 questions right. show all your steps.

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98 points in total! 1 question! no calculator! show

WendigoMeow

10.03.2022

0.2460 is the required probability.

Step-by-step explanation:

We have been given binomial model

n is the total number of questions which is 10

x is the questions taken out to be which is 5

p is the probability of right answers which is : $\frac{5}{10}=\frac{1}{2}$

q is the probability of the false answers which is : $1-p=1-\frac{1}{2}=\frac{1}{2}$

We will use the model by substituting the values we get:

$P(x)=[\frac{n!}{x!(n-x)!}]p^xq^{n-x}$ on substituting the values we get:

$P(x)=[\frac{`10!}{5!(5)!}]\frac({1}{2})^5\cdot \frac{1}{2}^{5}$

$P(x)=\frac{10\cdot9\cdot8\cdot7\cdot6\cdot5!}{5!(5\cdot4\cdot3\cdot2\cdot1)}\cdot\frac{1}{2}^5\cdot\frac{1}{2}^5$

Cancel out the common terms from numerator and denominator we get:

$\frac{10\cdot9\cdot8\cdot7\cdot6}{5\cdot4\cdot3\cdot2}\frac{1}{32}\cdot\frac{1}{32}$

$\Rightarrow 14\cdot18\cdot\frac{1}{1024}$

$\Rightarrow \frac{14\cdot18}{1024}$

$\Rightarrow \frac{252}{1024}$

$\Rightarrow 0.2460$

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