Answer :
To determine which quiz the student should study for by comparing the probabilities of getting all questions correct, let's first find the probabilities for both subjects.
### History Quiz
\- Number of questions: [tex]\( 5 \)[/tex]
\- Probability of getting each question correct: [tex]\( 0.5 \)[/tex]
To find the probability of getting all [tex]\( 5 \)[/tex] questions correct in history, we need to raise the individual probability to the power of the number of questions:
[tex]\[ P(\text{all correct in history}) = 0.5^5 = \frac{1}{2^5} = \frac{1}{32} = 0.03125 \][/tex]
### Science Quiz
\- Number of questions: [tex]\( 3 \)[/tex]
\- Probability of getting each question correct: [tex]\( 0.25 \)[/tex]
To find the probability of getting all [tex]\( 3 \)[/tex] questions correct in science, we raise the individual probability to the power of the number of questions:
[tex]\[ P(\text{all correct in science}) = 0.25^3 = \left(\frac{1}{4}\right)^3 = \frac{1}{4^3} = \frac{1}{64} = 0.015625 \][/tex]
Now, we compare the probabilities:
\- Probability of getting all questions correct in history: [tex]\( 0.03125 \)[/tex]
\- Probability of getting all questions correct in science: [tex]\( 0.015625 \)[/tex]
Since the probability of getting all questions correct in science ([tex]\( 0.015625 \)[/tex]) is lower than the probability of getting all questions correct in history ([tex]\( 0.03125 \)[/tex]), the student is less likely to guess all questions correctly in the science quiz.
Therefore, the student should study for the quiz where the chances of guessing all questions correctly are lower, which is the science quiz.
Thus, the correct answer is:
[tex]\[ \text{B. Science, because the probability of getting all questions correct is 0.016, which is lower than for history (0.031).} \][/tex]
### History Quiz
\- Number of questions: [tex]\( 5 \)[/tex]
\- Probability of getting each question correct: [tex]\( 0.5 \)[/tex]
To find the probability of getting all [tex]\( 5 \)[/tex] questions correct in history, we need to raise the individual probability to the power of the number of questions:
[tex]\[ P(\text{all correct in history}) = 0.5^5 = \frac{1}{2^5} = \frac{1}{32} = 0.03125 \][/tex]
### Science Quiz
\- Number of questions: [tex]\( 3 \)[/tex]
\- Probability of getting each question correct: [tex]\( 0.25 \)[/tex]
To find the probability of getting all [tex]\( 3 \)[/tex] questions correct in science, we raise the individual probability to the power of the number of questions:
[tex]\[ P(\text{all correct in science}) = 0.25^3 = \left(\frac{1}{4}\right)^3 = \frac{1}{4^3} = \frac{1}{64} = 0.015625 \][/tex]
Now, we compare the probabilities:
\- Probability of getting all questions correct in history: [tex]\( 0.03125 \)[/tex]
\- Probability of getting all questions correct in science: [tex]\( 0.015625 \)[/tex]
Since the probability of getting all questions correct in science ([tex]\( 0.015625 \)[/tex]) is lower than the probability of getting all questions correct in history ([tex]\( 0.03125 \)[/tex]), the student is less likely to guess all questions correctly in the science quiz.
Therefore, the student should study for the quiz where the chances of guessing all questions correctly are lower, which is the science quiz.
Thus, the correct answer is:
[tex]\[ \text{B. Science, because the probability of getting all questions correct is 0.016, which is lower than for history (0.031).} \][/tex]
