数学代写|Math 130A Introduction to probability and stochastic porcesses Assignment 3

(1). If you roll a regular dice twice, what is the probability that the first outcome is 3 given that the sum of the outcomes is $8 ?$
(2). If you toss a fair coin twice, given that one of the outcome is head, what is the probability that the other outcome is head?
(3). If you roll a regular dice twice, what is the probability that one of the outcome is 6 given that the outcomes are different?

Suppose we have 12 women and 4 men. If we divide these 16 people into 4 groups of size $4,$ what is the probability that each group contains a man?

Suppose $n>m$. Let $\mathcal{S}$ be the collection of all functions from $[n]$ to $[\mathrm{m}]$. We randomly pick a function $f$ from $\mathcal{S}$. Define event $A=\{f$ is surjective $\}$.
(1). Find $\mathbb{P}[A]$
(2). What is the limit of $\mathbb{P}[A]$ as $n \rightarrow \infty$ and $m$ is fixed?

 Suppose we have a test for a certain disease. If a person has the disease, the test has $95 \%$ chance to be positive. If a person is healthy, the test has $5 \%$ chance to give a false positive result. Given that the test for a person is positive, what is the probability that the person has the disease?

Problem $5 .$ Let $A$ and $B$ be events such $0<\mathbb{P}[B]<1$ and $0<\mathbb{P}[A]<1$.
(1). Prove that $\mathbb{P}[A \mid B]>\mathbb{P}[A]$ if and only if $\mathbb{P}[B \mid A]>\mathbb{P}[B]$.
(2). Prove that $\mathbb{P}[A \mid B]>\mathbb{P}[A]$ if and only if $\mathbb{P}\left[A \mid B^{C}\right]<\mathbb{P}[A]$
(3). Let $S=\{1, \cdots, n\}$. If we pick two numbers from $S$ with replacement. Let $E=\{$ the first pick is an even number $\}, F=\{$ the sum of two numbers is even $\}$. When is $\mathbb{P}[F \mid E]<\mathbb{P}[F] ?$