To construct a binomial probability distribution for [tex]\( n = 6 \)[/tex] and [tex]\( p = 0.6 \)[/tex], we need to find the probabilities for each value of [tex]\( x \)[/tex] from [tex]\( 0 \)[/tex] to [tex]\( 6 \)[/tex].
The probabilities are given by:
[tex]\[
P(X = x) = \binom{n}{x} p^x (1 - p)^{n - x}
\][/tex]
For [tex]\( n = 6 \)[/tex] and [tex]\( p = 0.6 \)[/tex], we have the following probabilities for each [tex]\( x \)[/tex]:
[tex]\[
\begin{array}{c|l|l}
x & P(X = x) \\
\hline
0 & 0.0041 \\
\hline
1 & 0.0369 \\
\hline
2 & 0.1382 \\
\hline
3 & 0.2765 \\
\hline
4 & 0.3110 \\
\hline
5 & 0.1866 \\
\hline
6 & 0.0467 \\
\hline
\end{array}
\][/tex]
These probabilities have been rounded to four decimal places.
So, the binomial probability distribution is:
[tex]\[
\begin{array}{c|c}
x & P(X = x) \\
\hline
0 & 0.0041 \\
\hline
1 & 0.0369 \\
\hline
2 & 0.1382 \\
\hline
3 & 0.2765 \\
\hline
4 & 0.3110 \\
\hline
5 & 0.1866 \\
\hline
6 & 0.0467 \\
\hline
\end{array}
\][/tex]
