To determine the probability that the hens lay a total of two eggs, we need to analyze the given probability distribution table. The table shows the probabilities for the number of eggs laid, except for the probability of laying exactly two eggs, which is represented by a question mark '?'.
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
\text{Number of Eggs} & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
\text{Probability} & 0.02 & 0.03 & ? & 0.12 & 0.30 & 0.28 & 0.18 \\
\hline
\end{array}
\][/tex]
First, let's write down the given probabilities:
[tex]\[
\begin{aligned}
& P(X = 0) = 0.02, \\
& P(X = 1) = 0.03, \\
& P(X = 3) = 0.12, \\
& P(X = 4) = 0.30, \\
& P(X = 5) = 0.28, \\
& P(X = 6) = 0.18. \\
\end{aligned}
\][/tex]
Since the total probability for all possible outcomes must be equal to 1, we can use this to find the missing probability [tex]\(P(X = 2)\)[/tex]:
[tex]\[
P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) = 1
\][/tex]
Substituting the given values into the equation:
[tex]\[
0.02 + 0.03 + P(X = 2) + 0.12 + 0.30 + 0.28 + 0.18 = 1
\][/tex]
Adding up the known probabilities:
[tex]\[
0.02 + 0.03 + 0.12 + 0.30 + 0.28 + 0.18 = 0.93
\][/tex]
Now, solving for [tex]\(P(X = 2)\)[/tex]:
[tex]\[
0.93 + P(X = 2) = 1
\][/tex]
[tex]\[
P(X = 2) = 1 - 0.93
\][/tex]
[tex]\[
P(X = 2) = 0.07
\][/tex]
Thus, the probability that the hens lay a total of two eggs is [tex]\(0.07\)[/tex].
Comparing this result with the given answer choices:
- 0.05
- 0.07
- 0.12
- 0.93
The correct choice is [tex]\(0.07\)[/tex].
