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You have a 5-question multiple-choice test. Each question has four choices. You don’t know any of the answers. What is the experimental probability that you will guess exactly three out of five questions correctly?

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frika
Each question has four choices, then the probability to guess a correct answer is [tex]p= \frac{1}{4} [/tex] and the probability to select incorrect choice is  [tex]q=1-p=1- \frac{1}{4} = \frac{3}{4} [/tex].

You have a 5-question multiple-choice test, then n=5. The probability that you will guess exactly three out of five questions correctly is

[tex]C_n^kp^nq^{n-k}=C_5^3p^3q^{5-3}= \dfrac{5!}{3!\cdot 2!} \left( \dfrac{1}{4} \right)^3\left( \dfrac{3}{4} \right)^2=\dfrac{1\cdot 2\cdot 3\cdot 4\cdot 5}{1\cdot 2\cdot 3\cdot 1\cdot 2} \cdot \dfrac{9}{1024} =[/tex]
[tex]= \dfrac{90}{1024} = \dfrac{45}{512} [/tex].

Answer with explanation:

Number of Multiple Choice Question=5

Number of choices =4

Out of 4 choices one is correct Option.

Probability of Correct answer P(C→Correct)

  [tex]=\frac{\text{Total favorable Outcome}}{\text{Total Possible Outcome}}=\frac{1}{4}[/tex]

Probability of Incorrect answer P(IC→Incorrect Answer)

   [tex]1-\frac{1}{4}=\frac{3}{4}[/tex]

→Experimental Probability of guessing three out of five questions correctly

= 3 Correct +2 Incorrect

[tex]=_{3}^{5}\textrm{C}[P(C)]^3\times [P(I C)]^2\\\\=\frac{5!}{3!\times 2!}\times[\frac{1}{4}]^3 \times[\frac{3}{4}]^2\\\\=10 \times \frac{1}{64} \times \frac{9}{16}\\\\=\frac{90}{1024}\\\\=0.87890[/tex]

  =0.88(approx)

     

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