So I have some probability questions below, and I did the first 3 questions and was wondering if someone could confirm my answers for them. I'm not exactly sure how I would go about doing 4 and 5 and was hoping for some clarification. My friends are all getting different answers for the last 2 questions and I'm not sure how to do them. Any help would be appreciated!
We independently pick two random numbers R1 and R2 from the set {1,…,1000}. (Note: Independence means we may pick R1=R2.)
- What is the probability that R1 and R2 are both even?
Let P(A) equal the probability that R1 is even, and P(B) equal the probability that R2 is even.
P(A and B) = P(A) * P(B) = (1/2) * (1/2) = 1/4
- Suppose I tell you that at least one of R1 or R2 is even. What is the (conditional) probability that R1 and R2 are both even?
Let P(A) equal the probability that R1 is even, and P(B) equal the probability that R2 is even.
P(A) = 1/2 × 1/2 = 1/4
P(B) = 1 - P(A') = 1 - 1/4 = 3/4
P(A|B) = (1/4)/(3/4) = 1/3
- What is the probability that at least one of R1 or R2 is equal to 1000?
Let P(A) represent the probability that R1 is equal to 1000, and P(B) be the probability that R2 is equal to 1000.
P(A'and B')=P(A')× P(B')= 0.999 × 0.999 = 0.998001
P(A or B) = 1 - P(A' and B') = 1 - 0.998001 = 0.001999
Suppose I tell you that at least one of R1 or R2 is even. What is the probability that at least one of R1 or R2 is equal to 1000?
Suppose I tell you that at least one of R1 or R2 is even. What is the probability that at least one of R1 or R2 is equal to 999?
The answers to the first 3 seems correct but your definition of $A$ and $B$ in question 2 seem a bit off. For the question 4, recall that as long as $P(B) > 0$,
$$ P(A \mid B) = P(A \cap B) / P(B). $$
Your events are $B = \{\text{One of R1 or R2 is even}\}$ and $A = \{\text{One of R1 or R2 = 1000}\}$. Note that $A \subset B$ in this case, therefore $A \cap B = A$ and note you've already computed $P(A)$ and $P(B)$.
Question 5 is solved in the same way, just be careful of what $A \cap B$ is.