csat-suneung 2026 Q29

csat-suneung · South-Korea · csat__math 4 marks Approximating Binomial to Normal Distribution
For a natural number $a$ not exceeding 6, a trial is performed using one die and one coin.
Roll the die once. If the result is less than or equal to $a$, flip the coin 5 times and record the number of heads. If the result is greater than $a$, flip the coin 3 times and record the number of heads.
This trial is repeated 19200 times, and let $X$ be the number of times the recorded number is 3. When $\mathrm { E } ( X ) = 4800$, find the value of $\mathrm { P } ( X \leq 4800 + 30 a )$ using the standard normal distribution table below, which equals $k$.
$z$$\mathrm { P } ( 0 \leq Z \leq z )$
0.50.191
1.00.341
1.50.433
2.00.477
2.50.494
3.00.499

Find the value of $1000 \times k$. [4 points] 
For a natural number $a$ not exceeding 6, a trial is performed using one die and one coin.

Roll the die once.
If the result is less than or equal to $a$, flip the coin 5 times and record the number of heads.
If the result is greater than $a$, flip the coin 3 times and record the number of heads.

This trial is repeated 19200 times, and let $X$ be the number of times the recorded number is 3.
When $\mathrm { E } ( X ) = 4800$, find the value of $\mathrm { P } ( X \leq 4800 + 30 a )$ using the standard normal distribution table below, which equals $k$.

\begin{center}
\begin{tabular}{ | c | c | }
\hline
$z$ & $\mathrm { P } ( 0 \leq Z \leq z )$ \\
\hline
0.5 & 0.191 \\
\hline
1.0 & 0.341 \\
\hline
1.5 & 0.433 \\
\hline
2.0 & 0.477 \\
\hline
2.5 & 0.494 \\
\hline
3.0 & 0.499 \\
\hline
\end{tabular}
\end{center}

Find the value of $1000 \times k$. [4 points]
Paper Questions
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