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\begin{tabular}{|c|c|c|}
\hline Place & Is a city & Is in North America \\
\hline Rome & [tex]$V$[/tex] & \\
\hline Tokyo & [tex]$V$[/tex] & \\
\hline Houston & [tex]$V$[/tex] & [tex]$\checkmark$[/tex] \\
\hline Peru & & \\
\hline Miami & [tex]$\checkmark$[/tex] & [tex]$\checkmark$[/tex] \\
\hline Toronto & [tex]$\checkmark$[/tex] & [tex]$\checkmark$[/tex] \\
\hline Canada & & [tex]$\checkmark$[/tex] \\
\hline
\end{tabular}

A place from this table is chosen at random. Let event [tex]$A=$[/tex] The place is a city.

What is [tex]$P\left(A^C\right)$[/tex]?
A. [tex]$\frac{5}{7}$[/tex]
B. [tex]$\frac{3}{7}$[/tex]
C. [tex]$\frac{2}{7}$[/tex]
D. [tex]$\frac{4}{7}$[/tex]



Answer :

GinnyAnswer
Alright, let's work through this question step by step.

1. Identify Total Number of Places:
From the table, there are 7 places listed.

2. Count the Number of Cities:
The places which are cities (marked with V) from the table are:
- Rome
- Tokyo
- Houston
- Miami
- Toronto

Thus, the total number of cities is 5.

3. Calculate Probability of Chosen Place Being a City:
The probability \( P(A) \) that a randomly chosen place is a city is given by:
[tex]\[ P(A) = \frac{\text{Number of cities}}{\text{Total number of places}} = \frac{5}{7} \][/tex]

4. Calculate Probability of Complement Event (Not a City):
The complementary event \( A^C \) is the event that the chosen place is not a city. The probability \( P(A^C) \) can be calculated as:
[tex]\[ P(A^C) = 1 - P(A) \][/tex]
Therefore:
[tex]\[ P(A^C) = 1 - \frac{5}{7} = \frac{7}{7} - \frac{5}{7} = \frac{2}{7} \][/tex]

So, the probability \( P(A^C) \), which is the probability that the chosen place is not a city, is \(\frac{2}{7}\).

Therefore, the correct answer is:
C. [tex]\(\frac{2}{7}\)[/tex].

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