\begin{tabular}{|c|c|c|}
\hline
Place & Is a city & Is in North America \\
\hline
India & & \\
\hline
Tokyo & v & \\
\hline
Houston & v & \\
\hline
Peru & & \\
\hline
New York & v & \\
\hline
Tijuana & v & \\
\hline
Canada & & v \\
\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]$\frac{2}{7}$[/tex]



Answer :

To determine [tex]\( P(A^C) \)[/tex], we need to understand both event [tex]\( A \)[/tex] and its complement [tex]\( A^C \)[/tex].

1. Identify Event [tex]\( A \)[/tex]:
- Event [tex]\( A \)[/tex] is defined as the place chosen being a city.

2. Count the Total Number of Places:
- There are a total of 7 places listed in the table.

3. Identify the Places that are Cities:
- The places that are cities from the table are Tokyo, Houston, New York, and Tijuana. Thus, there are 4 cities.

4. Determine the Complement Event [tex]\( A^C \)[/tex]:
- The complement of the event [tex]\( A \)[/tex] (denoted as [tex]\( A^C \)[/tex]) is the event that the place chosen is not a city.

5. Count the Number of Non-City Places:
- Total places = 7
- Number of cities = 4
- Number of non-cities = Total places - Number of cities
- Number of non-cities = 7 - 4 = 3

6. Calculate the Probability [tex]\( P(A^C) \)[/tex]:
- Probability of [tex]\( A^C \)[/tex] is the ratio of non-city places to the total number of places.
- [tex]\( P(A^C) = \frac{\text{Number of non-cities}}{\text{Total number of places}} \)[/tex]
- [tex]\( P(A^C) = \frac{3}{7} \)[/tex]

Therefore, the probability that a randomly chosen place is not a city is [tex]\( \frac{3}{7} \)[/tex].

Converting the fraction to a decimal, we get [tex]\( \frac{3}{7} \approx 0.42857142857142855 \)[/tex].

So, the probability [tex]\( P(A^C) \)[/tex] is approximately 0.42857142857142855.