Four students, Rick, Robin, Ryu, and Roy
are registered for the same class and attend inde
pendently of one
another, Rick 96.5% of the time, Robin 93.3% of the time, Ryu
8
8.2
% of the time,
and
Roy
77.9
% of the time.
What is the probability that on any given day
a)
at least three of them will be in class?
b)
none of them will be in class?
To find the probability that at least three of the four students will be in class on any given day, we need to calculate the probability for each possible scenario.
a) At least three students in class
To solve this, we need to calculate the probability of different combinations:
1. Rick, Robin, and Ryu are in class, while Roy is absent.
Probability = Rick's probability * Robin's probability * Ryu's probability * (1 - Roy's probability)
= 0.965 * 0.933 * 0.882 * (1 - 0.779)
2. Rick, Robin, and Roy are in class, while Ryu is absent.
Probability = Rick's probability * Robin's probability * (1 - Ryu's probability) * Roy's probability
= 0.965 * 0.933 * (1 - 0.882) * 0.779
3. Rick, Ryu, and Roy are in class, while Robin is absent.
Probability = Rick's probability * (1 - Robin's probability) * Ryu's probability * Roy's probability
= 0.965 * (1 - 0.933) * 0.882 * 0.779
4. Robin, Ryu, and Roy are in class, while Rick is absent.
Probability = (1 - Rick's probability) * Robin's probability * Ryu's probability * Roy's probability
= (1 - 0.965) * 0.933 * 0.882 * 0.779
Now, we need to add up the probabilities of all these scenarios to get the total probability of at least three of them being in class:
Total Probability = Probability of scenario 1 + Probability of scenario 2 + Probability of scenario 3 + Probability of scenario 4
b) None of them in class
To find the probability that none of the students will be in class, we need to calculate the probability that each student is absent:
Probability = (1 - Rick's probability) * (1 - Robin's probability) * (1 - Ryu's probability) * (1 - Roy's probability)
Now, let's calculate the probabilities for both scenarios.