To determine the probability that Karen answers the first three questions correctly and the last three questions incorrectly on a true-false quiz of six questions, we can follow these steps:
1. Calculate the Probability of Answering a Single Question Correctly or Incorrectly:
Since the quiz is true-false, there are only two possible outcomes for each question – "true" or "false". Therefore, the probability of guessing any single question correctly is:
[tex]\[
p_\text{correct} = \frac{1}{2}
\][/tex]
Similarly, the probability of guessing any single question incorrectly is:
[tex]\[
p_\text{incorrect} = \frac{1}{2}
\][/tex]
2. Probability of Answering the First Three Questions Correctly:
The probability of answering the first question correctly is [tex]\( \frac{1}{2} \)[/tex]. Since each question is independent of the others, the probability that Karen guesses the first three questions correctly is the product of the individual probabilities:
[tex]\[
p_\text{first three correct} = \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^3 = \frac{1}{8}
\][/tex]
3. Probability of Answering the Last Three Questions Incorrectly:
Similarly, the probability of answering the last question incorrectly is [tex]\( \frac{1}{2} \)[/tex]. The probability that Karen guesses the last three questions incorrectly is:
[tex]\[
p_\text{last three incorrect} = \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^3 = \frac{1}{8}
\][/tex]
4. Total Probability of the Specified Event:
To find the total probability that Karen answers the first three questions correctly and the last three questions incorrectly, we multiply the probabilities of these two independent events:
[tex]\[
p_\text{total} = p_\text{first three correct} \times p_\text{last three incorrect} = \frac{1}{8} \times \frac{1}{8} = \frac{1}{64}
\][/tex]
So, the probability that Karen answers the first three questions correctly and the last three questions incorrectly is:
[tex]\[
\boxed{\frac{1}{64}}
\][/tex]
