# Tutorial 2 Notes

### Gubner Probability Textbook p.48, Question 6

Sketch the following subsets of the x-y plane.

### Gubner p.48 Question 10

#### Explain why the function $f(- \infty, \infty ) \rightarrow [0, \infty )$ with $f(x)=x^3$ is not well defined.

Here, the range is $(- \infty, \infty )$ because the range is not a subset of the co-domain $[0, \infty)$, this function is not well defined.

### Gubner p.50 Question 13

#### If $f : x \rightarrow y$, show that inverse images preserve the following set operations. (a) If $B \subset Y$, show that $f^{-1} (B^C) = f^{-1}(B)^C$ (b+c) Please look it up in the book.

(a)

$$x \in f^{-1} (B^C) \Leftrightarrow f(x) \in B^C$$ $$f^{-1}(B) = \{ x \in X : f(x) \in B \}$$ $$\Leftrightarrow f(x) \not\in B \Leftrightarrow x \not\in f^{-1}(B) \Leftrightarrow x \in (f^{-1}(B))^C$$ $$f^{-1}(B^C) = (f^{-1}(B))^C$$

(b)

$$x \in f^{-1}(\bigcup_{n=1}^{\infty} B_n) \Leftrightarrow f(x) \in \bigcup_{n=1}^{\infty}B_n$$ $$\Leftrightarrow f(x) \in B_n, \text{for some n}$$ $$\Leftrightarrow x \in f^{-1} (B_n), \text{for some n}$$ $$\Leftrightarrow x \in \bigcup_{n=1}^{\infty} f^{-1}(B_n)$$

(c)

$$x \in f^{-1} (\bigcap_{n=1}^{\infty} B_n) \Leftrightarrow f(x) \in \bigcap_{n=1}^{\infty} B_n$$ $$\Leftrightarrow f(x) \in B_n, \forall n$$ $$\Leftrightarrow x \in f^{-1}(B_n), \forall n$$ $$\Leftrightarrow x \in \bigcup_{n=1}^{\infty} f^{-1}(B_n)$$

### p.50 Question 24

#### A collection plastic letters a-z are mixed in a jar. Two letters are drawn one after the other. What is the probabilty of drawing a vowel (a, e, i, o, u) and a constonant in any order? Two vowels, in any order? Specify the sample space $\Omega$ and probability for the events just described.

(a) Sample space, $\Omega = { (i, j) 1 \le i \le 26, 1 \le j \le 26, i \ne j }$ and $\| \Omega \| = 26 * 25 = 650$

We define:

Because, wehn the sample space is finite, $% $ and the outcomes (the elements of $\Omega$ ) are qually likely to occur, then for any $A < \Omega$. $P(A) = \frac{\| A \|}{\| \Omega \|}$

Two vowels?

### p.53, Question 29

#### Let A and B be events for which $P(A)$, $P(B)$, and $P(A \cup B )$ are known. Express the following in terms of these probabilities. (a) $P(A \cap B)$ (b) $P( A \cup B^C )$ (c) $P(B \cup (A \cap B^C ) )$ (d) $P(A \cap B^C)$

(a)

The inclusive-exlcusive principle states that $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Written on January 19, 2015