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At a game show, there are three doors marked 1, 2, and 3. A contestant is allowed to select one of three keys labeled A, B, and C and then use it to open one of the doors. The sample space showing the ways in which a contestant can select a key and a door is represented in the table below.

\begin{tabular}{|c|c|c|c|c|}
\hline & & \multicolumn{3}{|c|}{ Door } \\
\hline & & 1 & 2 & 3 \\
\hline \multirow{4}{*}{ Key } & A & A-1 & & A-3 \\
\cline { 3 - 5 } & B & B-1 & B-2 & B-3 \\
\cline { 2 - 5 } & C & C-1 & C-2 & C-3 \\
\hline
\end{tabular}

What is the missing term in the table? If there were two doors instead of three doors, what would the sample size be?

A. The missing term is A-2. The sample size would be 6.
B. The missing term is C-2. The sample size would be 8.
C. The missing term is C-2. The sample size would be 9.
D. The missing term is B-2. The sample size would be 12.



Answer :

GinnyAnswer
To solve this question, let's break down the problem and analyze each part step-by-step.

1. Understanding the Table:
- We have three keys: A, B, and C.
- We have three doors: 1, 2, and 3.
- The table lists combinations of keys and doors in the form "X-Y", where X is the key and Y is the door number.

2. Initial Table Analysis:
- The table is partially filled. We are given:
- A -> A-1, (missing), A-3
- B -> B-1, B-2, B-3
- C -> C-1, C-2, C-3

- From this, we can see that each key should match with each door once. Therefore, the missing term in the table where Key A intersects with Door 2 should be "A-2".

3. Calculating the Initial Sample Size:
- In the initial scenario, we have 3 keys and 3 doors.
- The sample size can be calculated by multiplying the number of keys by the number of doors:
[tex]\[ \text{Initial Sample Size} = 3 \times 3 = 9 \][/tex]

4. Calculating the Reduced Sample Size:
- If we reduce the number of doors to 2 while keeping the number of keys the same (3 keys: A, B, C; 2 doors: 1, 2), the new sample size can be calculated as:
[tex]\[ \text{Reduced Sample Size} = 3 \times 2 = 6 \][/tex]

5. Conclusion:
- The missing term in the table is "A-2".
- If there were only two doors, the sample size would be 6.

So, the correct answer is:

A. The missing term is A-2. The sample size would be 6.

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