Sure, let's solve this problem step-by-step.
1. Identify the given probabilities:
- [tex]\( P(\text{getting exactly 7 correct}) = 0.0031 \)[/tex]
- [tex]\( P(\text{getting exactly 8 correct}) = 0.000386 \)[/tex]
- [tex]\( P(\text{getting exactly 9 correct}) = 2.86 \times 10^{-5} \)[/tex]
- [tex]\( P(\text{getting exactly 10 correct}) = 9.54 \times 10^{-7} \)[/tex]
2. Understand what we need to find:
We are asked to compute the total probability that Peter will pass the quiz. Peter passes the quiz if he gets at least 7 questions correct. This means we need to sum the probabilities of getting exactly 7, exactly 8, exactly 9, and exactly 10 questions correct.
3. Add the individual probabilities:
[tex]\[
\begin{aligned}
P(\text{passing the quiz}) &= P(\text{getting exactly 7 correct}) \\
&\quad+ P(\text{getting exactly 8 correct}) \\
&\quad+ P(\text{getting exactly 9 correct}) \\
&\quad+ P(\text{getting exactly 10 correct}) \\
&= 0.0031 + 0.000386 + 2.86 \times 10^{-5} + 9.54 \times 10^{-7}
\end{aligned}
\][/tex]
4. Calculate the sum:
[tex]\[
0.0031 + 0.000386 + 0.0000286 + 0.000000954 = 0.003515554
\][/tex]
5. Conclusion:
The total probability that Peter will pass the quiz by getting at least 7 questions correct is [tex]\(0.003515554\)[/tex].
So, the correct answer is approximately:
[tex]\[ \boxed{0.0035} \][/tex]
