Three students, Chris, Vlad, and Steve, are registered for the same class and attend
independently of each other. If Chris is in class 80% of the time, Vlad is in class 85%
of the time, and Steve is in class 90% of the time, what is the probability that at least
two of them are in class on a randomly selected day?
0.941
To find the probability that at least two of them are in class on a randomly selected day, we can calculate the probability of the complement event (i.e., the event that none or only one of them is in class) and subtract it from 1.
Let's start by calculating the probability that Chris is not in class:
P(Chris not in class) = 1 - P(Chris in class) = 1 - 0.80 = 0.20
Similarly, the probabilities that Vlad and Steve are not in class are:
P(Vlad not in class) = 1 - P(Vlad in class) = 1 - 0.85 = 0.15
P(Steve not in class) = 1 - P(Steve in class) = 1 - 0.90 = 0.10
Now, let's consider the four possible cases:
1. None of them are in class:
P(None in class) = P(Chris not in class) * P(Vlad not in class) * P(Steve not in class)
= 0.20 * 0.15 * 0.10
2. Only Chris is in class:
P(Only Chris in class) = P(Chris in class) * P(Vlad not in class) * P(Steve not in class)
= 0.80 * 0.15 * 0.10
3. Only Vlad is in class:
P(Only Vlad in class) = P(Chris not in class) * P(Vlad in class) * P(Steve not in class)
= 0.20 * 0.85 * 0.10
4. Only Steve is in class:
P(Only Steve in class) = P(Chris not in class) * P(Vlad not in class) * P(Steve in class)
= 0.20 * 0.15 * 0.90
Now, we can calculate the probability that at least two of them are in class:
P(At least two in class) = 1 - P(None in class) - P(Only Chris in class) - P(Only Vlad in class) - P(Only Steve in class)
Hope this explanation helps! Let me know if you have any further questions.