Calculating probability of three independent events.

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Bigwig

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I'm trying to teach myself probability, and I keep getting stuck on the problem sets. I'm using DeGroot ​Probability and Statistics 2nd Ed. This problem is 1.13 #10

Three students, A, B, and C are enrolled in the same class. Suppose that A attends class 30 percent of the time, B attends class 50 percent of the time, and C attends class 80 percent of the time. If these students attend class independently of each other, what is (a) the probability that at least one of them will be in class on a particular day, and (b) the probability that exactly one of them will be in class on a particular day?

(1) I know that for independent events, the probability of the intersection of any number of events is simply the multiplication of their probabilities. For example Pr(ABC)= P(A)Pr(B)P(C)

I'm fairly certain that the situations I need to consider are that no one shows up, any one of the three students shows up, any two of the students shows up, and all three show up. (I believe this makes 8 possible situations) Using the rule in (1), I can compute the probability for any one of these situations. For example, the probability that all three show up is .12, just B and C .4, etc.

I do not know how to get from this to the answers to (a) and (b). I thought it would somehow involve using the formula for a union of events, but I can't see a way to make that work.

Can someone help me conceptualize this problem correctly?
 
Bigwig said:
Three students, A, B, and C are enrolled in the same class. Suppose that A attends class 30 percent of the time, B attends class 50 percent of the time, and C attends class 80 percent of the time. If these students attend class independently of each other, what is (a) the probability that at least one of them will be in class on a particular day, and (b) the probability that exactly one of them will be in class on a particular day?
Click to expand...
Use the notation \(\displaystyle \overline{A}\) for the complement of event \(\displaystyle A\) (not \(\displaystyle A\)).
So \(\displaystyle \mathcal{P}(\overline{A})=0.7\).

At least one of the three present is \(\displaystyle 1-\mathcal{P}(\overline{A})\mathcal{P}(\overline{B}) \mathcal{P}(\overline{C})\)
 
Bigwig said:
I'm trying to teach myself probability, and I keep getting stuck on the problem sets. I'm using DeGroot ​Probability and Statistics 2nd Ed. This problem is 1.13 #10

Three students, A, B, and C are enrolled in the same class. Suppose that A attends class 30 percent of the time, B attends class 50 percent of the time, and C attends class 80 percent of the time. If these students attend class independently of each other, what is (a) the probability that at least one of them will be in class on a particular day, and (b) the probability that exactly one of them will be in class on a particular day?

(1) I know that for independent events, the probability of the intersection of any number of events is simply the multiplication of their probabilities. For example Pr(ABC)= P(A)Pr(B)P(C)

I'm fairly certain that the situations I need to consider are that no one shows up, any one of the three students shows up, any two of the students shows up, and all three show up. (I believe this makes 8 possible situations) Using the rule in (1), I can compute the probability for any one of these situations. For example, the probability that all three show up is .12, just B and C .4, etc.

I do not know how to get from this to the answers to (a) and (b). I thought it would somehow involve using the formula for a union of events, but I can't see a way to make that work.

Can someone help me conceptualize this problem correctly?
Click to expand...
There are frequently a number of ways to solve a problem in probability theory.

Many problems ask for the probability of at least x "successes" out of n trials, \(\displaystyle 1 \le x \le n.\)

One way: \(\displaystyle \displaystyle \sum_{i=1}^xP(i).\)

A different way : \(\displaystyle \displaystyle 1 - \sum_{i=0}^{x - 1}P(i).\)

Do whichever involves less computation.

A similar logic applies to questions with "at most" in them.
 
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