2.4 Product rule

Discrete probability law


If the sample space \(\Omega\) consists of a finite number of possible outcomes,

then the probability of any event \(\{s_1, s_2, \cdots, s_k \}\)

is the sum of the probabilities of its elements.


\[\text{P}(\{s_1, s_2, \cdots, s_k \}) = \text{P}(s_1) + \text{P}(s_2) + \cdots + \text{P}(s_k) \]

Discrete uniform probability law

If all outcomes in the sample space \(\Omega\) are equally likely,

then the probability of any event A is given by:

\[ \text{P}(A) = \frac{\text{number of outcomes in } A}{\text{number of outcomes in } \Omega} \]


For convenience, we use \(|A|\) to denote the number of elements in set \(A\) (termed the cardinality of set \(A\))

\[ \text{P}(A) = \frac{|A|}{|\Omega|} \]

Equally likely outcomes

In many experiments, it is reasonable to assume equal probabilities to all simple events.


Roll a fair six-sided die 🎲

\[ \text{P}(⚀) = \text{P}(⚁) = \text{P}(⚂) = \text{P}(⚃) = \text{P}(⚄) = \text{P}(⚅) = \frac{1}{6} \]

\[\text{P}(\text{odd number})=\text{P}\big(\{⚀, ⚂, ⚄\}\big)=\frac{3}{6}\]

Roll two dice 🎲 🎲

  • What is the probability to roll a double?
  • What is the probability that the total is over 7?

Draw a card from a well-shuffled deck of 52 cards.

What is the chance to draw an ace?

\[ \text{P}(\text{Aces}) = \text{P}(\{🃑, 🃁, 🂱, 🂡\}) = \frac{4}{52}= \frac{1}{13} \]

Draw a card from a well-shuffled deck of 52 cards.

What is the chance to draw a spade?

\[ \text{P}(\text{spades}) = \text{P}(\{🂡, 🂢, \cdots, 🂮\})= \frac{13}{52}= \frac{1}{4} \]

Roll a die twice. How many pairs of numbers are possible?


The product rule

Suppose that

  • there are \(\color{red}{n_1}\) possible results at the first stage
  • for every possible result at the first stage, there are \(\color{green}{n_2}\) possible results at the second stage

Then the total number of possible results is \(\color{red}{n_1} \times \color{green}{n_2}\)

The product rule: four stages

General form of the product rule with \(r\)-stages


If for any sequence of possible results at the first \((i-1)\) stages, there are \(n_i\) possible results at the \(i\)-th stage

then the total number of possible results in a \(r\)-stage process is

\[n_1 \times n_2 \times \cdots \times n_r\]

Exercise

How many iPad Pro variants does Apple need to make?

  • Screen size: 11-inch, 13-inch
  • Color: Space gray, Silver
  • Storage: 256GB, 512GB, 1TB, 2TB
  • Display glass: Standard, Nano-texture
  • Connectivity: Wi-Fi, Wi-Fi + Cellular

Curse of dimensionality

Exercise

How many unique MI license plate numbers are there?

Rules

  • 7 characters
  • Only uppercase letters (A-Z) and numbers (0-9)
  • Letter O is not allowed.
  • Spaces are allowed and counted as characters.

Exercise


How many subsets does a \(n\)-element set have?

\[ \{s_1, s_2, \cdots, s_n\} \]