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Sagot :
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."
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."
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