91Ó°ÊÓ

If \(P(A)>0, P(B)>0,\) and \(P(A)

Suppose that \(A \subset B\) and that \(P(A)>0\) and \(P(B)>0 .\) Show that \(P(B | A)=1\) and \(P(A | B)=P(A) / P(B)\).

If \(A\) and \(B\) are mutually exclusive events and \(P(B)>0\), show that $$P(A | A \cup B)=\frac{P(A)}{P(A)+P(B)}$$.

If \(A_{1}, A_{2},\) and \(A_{3}\) are three events and \(P\left(A_{1} \cap A_{2}\right)=P\left(A_{1} \cap A_{3}\right) \neq 0\) but \(P\left(A_{2} \cap A_{3}\right)=0,\) show that \(P\left(\text { at least one } A_{i}\right)=P\left(A_{1}\right)+P\left(A_{2}\right)+P\left(A_{3}\right)-2 P\left(A_{1} \cap A_{2}\right)\)

If \(A\) and \(B\) are independent events, show that \(A\) and \(\bar{B}\) are also independent. Are \(\bar{A}\) and \(\bar{B}\) independent?

Suppose that \(A\) and \(B\) are two events such that \(P(A)=.8\) and \(P(B)=.7\). a. Is it possible that \(P(A \cap B)=.1 ?\) Why or why not? b. What is the smallest possible value for \(P(A \cap B) ?\) c. Is it possible that \(P(A \cap B)=.77 ?\) Why or why not? d. What is the largest possible value for \(P(A \cap B) ?\)

Suppose that \(A\) and \(B\) are two events such that \(P(A)+P(B)>1\). a. What is the smallest possible value for \(P(A \cap B) ?\) b. What is the largest possible value for \(P(A \cap B) ?\)

\- Suppose that \(A\) and \(B\) are two events such that \(P(A)=.6\) and \(P(B)=.3\). a. Is it possible that \(P(A \cap B)=.1 ?\) Why or why not? b. What is the smallest possible value for \(P(A \cap B) ?\) c. Is it possible that \(P(A \cap B)=.7 ?\) Why or why not? d. What is the largest possible value for \(P(A \cap B) ?\)

Suppose that \(A\) and \(B\) are two events such that \(P(A)+P(B)<1\). a. What is the smallest possible value for \(P(A \cap B) ?\) b. What is the largest possible value for \(P(A \cap B) ?\)

Suppose that there is a 1 in 50 chance of injury on a single skydiving attempt. a. If we assume that the outcomes of different jumps are independent, what is the probability that a skydiver is injured if she jumps twice? b. A friend claims if there is a 1 in 50 chance of injury on a single jump then there is a \(100 \%\) chance of injury if a skydiver jumps 50 times. Is your friend correct? Why?