Respuesta :
The probability of getting the correct answer on one question is 1/2.
There are six questions, then (1/2)*(1/2)*(1/2)*(1/2)*(1/2)*(1/2) = (1/2)⁶ = 1/64
There are six questions, then (1/2)*(1/2)*(1/2)*(1/2)*(1/2)*(1/2) = (1/2)⁶ = 1/64
The probability of getting all six questions correct is [tex]\boxed{\dfrac{1}{{64}}}[/tex].
Further Explanation:
Probability can be defined as the ratio of favorable number of outcomes to the total number of outcomes.
Given:
There are [tex]6[/tex] true-false questions in an exam. A student answers all [tex]6[/tex] questions by guess.
Concept used:
The probability [tex]P\left( E \right)[/tex] of any event [tex]E[/tex] can be calculated as,
[tex]P\left( E \right) = \dfrac{{n\left( E \right)}}{{n\left( S \right)}}[/tex]
Here, [tex]n\left( E \right)[/tex] is the total number of elements in event [tex]E[/tex] and [tex]n\left( S \right)[/tex] is the number of element in sample space of an experiment.
Calculation:
The sample space is the total possible outcomes in an experiment.
Consider [tex]n\left( S \right)[/tex] as the number of element in sample space [tex]S[/tex].
The possible outcomes in sample space [tex]S[/tex] are either true or false.
Therefore, the number of element in sample space [tex]S[/tex] is,
[tex]n\left( S \right) = 2[/tex]
Consider [tex]A[/tex] as the event that the answer of first question is true, [tex]n\left( A \right)[/tex] as the number of ways that the answer of first question is true and [tex]P\left( A \right)[/tex] as the probability of event [tex]A[/tex].
The number of ways that the answer of first question is true can be calculated as,
[tex]n\left( A \right) = 1[/tex]
To obtain the probability [tex]P\left( A \right)[/tex], replace [tex]A[/tex] by [tex]E[/tex] in the equation [tex]P\left( E \right) = \dfrac{{n\left( E \right)}}{{n\left( S \right)}}[/tex] as,
[tex]P\left( A \right) = \dfrac{{n\left( A \right)}}{{n\left( S \right)}}[/tex]
Substitute [tex]1[/tex] for [tex]n\left( A \right)[/tex] and [tex]2[/tex] for [tex]n\left( S \right)[/tex] in the equation [tex]P\left( A \right) = \dfrac{{n\left( A \right)}}{{n\left( S \right)}}[/tex] to obtain the probability .
[tex]P\left( A \right) = \dfrac{1}{2}[/tex]
Now, the probability that all six questions are correct can be obtained as,
[tex]\begin{aligned}{\text{Probability}} &= \left( {\frac{1}{2}} \right) \times \left( {\frac{1}{2}} \right) \times \left( {\frac{1}{2}} \right) \times \left( {\frac{1}{2}} \right) \times \left( {\frac{1}{2}} \right) \times \left( {\frac{1}{2}} \right) \\ &= {\left( {\frac{1}{2}} \right)^6} \\&= \frac{1}{{64}} \\\end{aligned}[/tex]
The probability of getting all six questions correct is [tex]\boxed{\frac{1}{{64}}}[/tex].
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Probability
Keywords: Probability, outcome, total number of outcomes, ratio, favorable number of outcomes, six, six questions, correct, false, true, guess, consist, test, answer.
