Lecture 8

Qi Wang, Department of Statistics

Sep 7, 2018

- The first homework is due
**NOW** - The second quiz will be on this
**Wednesday(Sep 12)**

If there are a ways of doing something and b ways of doing another thing,
then there are a ยท b ways of performing both actions.

Assuming Mary has 6 pairs of shoes, 10 different tops, 8 different bottoms and 4 different jackets.

- How many different outfits can she wear?
- Mary has a job interview and she wants to decide what to wear. Of all her clothes, Mary has 2 pairs of shoes, 3 tops, 2 bottoms and 2 jackets that are appropriate for an interview. She randomly picks what to wear for the interview among all her possible outfits, what is the probability that s he wears an interview-appropriate outfit?

lllinois license plates consist of 4 digits followed by 2 letters. Whereas, in Ohio, license plates start with 3 letters and end with 4 digits. Assume all letters are upper case.(note: the license plate scheme described may not reflect the current Illinois or Ohio license plates)

- For each state, how many possible license plates are there?
- How many possible license plates are there for each state with no digit or letter repeating?
- How many possible license plates are there with at least 1 vowel?
- How many possible license plates are there with at least one vowel or at least one 3?
- What is the probability that the license plate will have at least one vowel?

- Factorial Notation: $k!$ means multiple the positive integer $k$ by $k - 1, k-2, \cdots, $ until 1 \begin{align} k! =& k\times (k-1) \times \cdots \times 2 \times 1 \\ 6! =& 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \end{align}
- Permutation: Ordered arrangement of r distinct objects from a set of n objects. $$_nP_r = P_r^n = \frac{n!}{(n - r)!}$$ $$_5P_2 = P_2^5 = \frac{5!}{(5 - 2)!} = \frac{5!}{3!} = 20$$

Suppose Krannert only allows 5 spaces for a password to Portals. Suppose further you are only allowed to use a number or a letter, but the system is not case sensitive.

- How many possible passwords are there?
- What is the probability that you do not have a 9 in the first position?
- What is the probability that all 5 spaces are odd numbers? What if you cannot have a 9 in the first space?
- What is the probability that a password does not repeat any characters?
- What is the probability that the first space is a letter?
- What is the probability that the $4_{th}$ space is an even number?
- What is the probability that the last two spaces are vowels, if repeats are allowed? If repeats are not allowed?
- What is the probability that the password has at least one letter?