Given the information in the table, which event is the least likely?

Skateboard Ownership at Seth's High School

| | Freshmen | Non-Freshmen | TOTAL |
|------------------------|----------|--------------|-------|
| Own skateboard | 40 | 110 | 150 |
| Does not own skateboard| 210 | 840 | 1,050 |
| TOTAL | 250 | 950 | 1,200 |

A. A randomly selected student who is a freshman owns a skateboard.

B. A randomly selected student who does not own a skateboard is not a freshman.

C. A randomly selected student who owns a skateboard is a freshman.

D. A randomly selected student who does not own a skateboard is a freshman.



Answer :

Let's analyze the given data step-by-step and calculate the required probabilities to determine which event is the least likely.

Firstly, let’s break down the data from the table:

[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{Freshmen} & \text{Non-Freshmen} & \text{TOTAL} \\ \hline \text{Own skateboard} & 40 & 110 & 150 \\ \hline \begin{array}{c} \text{Does not own} \\ \text{skateboard} \end{array} & 210 & 840 & 1050 \\ \hline \text{TOTAL} & 250 & 950 & 1200 \\ \hline \end{array} \][/tex]

Given the four events, we will calculate the probability of each event occurring:

1. Probability that a randomly selected freshman owns a skateboard:
[tex]\[ P(\text{Own skateboard} | \text{Freshman}) = \frac{\text{Number of freshmen who own a skateboard}}{\text{Total number of freshmen}} = \frac{40}{250} = 0.16 \][/tex]

2. Probability that a randomly selected student who does not own a skateboard is not a freshman:
[tex]\[ P(\text{Not Freshman} | \text{Does not own skateboard}) = \frac{\text{Number of non-freshmen who do not own a skateboard}}{\text{Total number of students who do not own a skateboard}} = \frac{840}{1050} = 0.8 \][/tex]

3. Probability that a randomly selected student who owns a skateboard is a freshman:
[tex]\[ P(\text{Freshman} | \text{Owns skateboard}) = \frac{\text{Number of freshmen who own a skateboard}}{\text{Total number of students who own a skateboard}} = \frac{40}{150} = 0.267 \][/tex]

4. Probability that a randomly selected student who does not own a skateboard is a freshman:
[tex]\[ P(\text{Freshman} | \text{Does not own skateboard}) = \frac{\text{Number of freshmen who do not own a skateboard}}{\text{Total number of students who do not own a skateboard}} = \frac{210}{1050} = 0.2 \][/tex]

Now, we have the probabilities for each event:
1. [tex]\( P(\text{Own skateboard} | \text{Freshman}) = 0.16 \)[/tex]
2. [tex]\( P(\text{Not Freshman} | \text{Does not own skateboard}) = 0.8 \)[/tex]
3. [tex]\( P(\text{Freshman} | \text{Owns skateboard}) = 0.267 \)[/tex]
4. [tex]\( P(\text{Freshman} | \text{Does not own skateboard}) = 0.2 \)[/tex]

The least likely event is the one with the smallest probability. Comparing the calculated probabilities:

[tex]\[ 0.16, 0.8, 0.267, 0.2 \][/tex]

The least likely event is:
[tex]\[ P(\text{Own skateboard} | \text{Freshman}) = 0.16 \][/tex]

Therefore, the event that is the least likely is "A randomly selected student who is a freshman owns a skateboard."