Question

after surveying students in her homeroom, katie decided to use a number cube to predict the probability that a randomly chosen student in her school planned to stay after school today for extracurricular activities. in katie’s simulation, a roll of 1 represents staying after school, while rolls of 2, 3, 4, 5, or 6 represent leaving school at the end of classes. according to katie’s model, what is the probability that three randomly chosen students will all stay after school for extracurricular activities today? \\(\frac{1}{729}\\) \\(\frac{1}{216}\\) \\(\frac{1}{18}\\) \\(\frac{1}{6}\\)

Explanation:

Step1: Find single student stay prob

A number cube has 6 outcomes. Only rolling a 1 means staying, so the probability for one student is $\frac{1}{6}$.

Step2: Calculate joint probability

For three independent students, multiply the individual probabilities: $\frac{1}{6} \times \frac{1}{6} \times \frac{1}{6}$

Step3: Compute final value

$\frac{1}{6} \times \frac{1}{6} \times \frac{1}{6} = \frac{1}{216}$

Answer:

$\frac{1}{216}$ (Option B)