Probability

Example: the chances of rolling a "4" with a die

Number of ways it can happen: 1 (there's only 1 face with a "4" on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability = 16

Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What's the probability that a blue marble gets picked at random?

Number of ways it can happen: 4 (there are 4 blues)

Total number of outcomes: 5 (there are 5 marbles in total)

So the probability = 45 = 0.8

Example: toss a coin 100 times, how many Heads will come up?

Probability says heads have a ½ chance, so we can expect 50 Heads.

But when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.

Example: Throwing dice

We can throw the dice again and again, so it is repeatable.

The set of possible results from any single throw is {1, 2, 3, 4, 5, 6}

Example: "6" is one of the outcomes of a throw of a die.

Example: I conducted a coin toss experiment. After 4 trials I got these results:

Three trials had the outcome "Head", and one trial had the outcome "Tail"

Example: Toss 1 coin once

We could get a Head or Tail:

Sample Space S = {H, T}

Example: Roll 1 die once

Sample Space S = {1, 2, 3, 4, 5, 6}

Example: Choosing a card from a deck

There are 52 cards in a deck (not including Jokers)

So the Sample Space is the set of all 52 possible cards:

Sample Space S = {Ace of Hearts, 2 of Hearts, and so on... }

Example: Deck of Cards

• the 5 of Clubs is a sample point
• the King of Hearts is a sample point

"King" isn't a sample point. There are 4 Kings, so that's 4 different sample points.

Example: Throwing dice

There are 6 different sample points in that sample space.

Example Events:

An event can be just one outcome:

• Getting a Tail when tossing a coin
• Rolling a "5" with a die

An event can also include more than one outcome:

• Choosing a "King" from a deck of cards (any of the 4 Kings)
• Rolling an "even number" (2, 4 or 6)

Example: Alex wants to see how many times a "double" comes up when throwing 2 dice.

The Sample Space is all possible Outcomes (36 Sample Points):

{1,1} {1,2} {1,3} {1,4} ...
...
... {6,3} {6,4} {6,5} {6,6}

The Event Alex is looking for is a "double", where both dice have the same number. It is made up of these 6 Sample Points:

{1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}

These are Alex's Results:

After 100 Trials, Alex has 19 "double" Events ... is that close to what you would expect?