Probability
How likely something is to happen.
Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.
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Tossing a Coin
When a coin is tossed, there are two possible outcomes:
We say that the probability of the coin landing H is ½.
And the probability of the coin landing T is ½.
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Throwing Dice
When a single dice is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6.
The probability of any one of them is 1/6.
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Probability
In general:
Probability of an event happening = Number of ways it can happenTotal number of outcomes
Example: the chances of rolling a "4" with a die
Number of ways it can happen: 1 (there is 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 is the probability that a blue marble gets picked?
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
Probability Line
We can show probability on a Probability Line:
Probability is always between 0 and 1
Probability is Just a Guide
Probability does not tell us exactly what will happen, it is just a guide
Example: toss a coin 100 times, how many Heads will come up?
Probability says that 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.
Words
Some words have special meaning in Probability:
Experiment or Trial: an action where the result is uncertain.
Tossing a coin, throwing dice, seeing what pizza people choose are all examples of experiments.
Sample Space: all the possible outcomes of an experiment
Example: choosing a card from a deck
There are 52 cards in a deck (not including Jokers)
So the Sample Space is all 52 possible cards: {Ace of Hearts, 2 of Hearts, etc... }
The Sample Space is made up of Sample Points:
Sample Point: just one of the possible outcomes
Example: Deck of Cards
- the 5 of Clubs is a sample point
- the King of Hearts is a sample point
"King" is not a sample point. As there are 4 Kings that is 4 different sample points.
Event: a single result of an experiment
Example Events:
- Getting a Tail when tossing a coin is an event
- Rolling a "5" is an event.
An event can include one or more possible outcomes:
- Choosing a "King" from a deck of cards (any of the 4 Kings) is an event
- Rolling an "even number" (2, 4 or 6) is also an event
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The Sample Space is all possible outcomes.
A Sample Point is just one possible outcome.
And an Event can be one or more of the possible outcomes.
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Hey, let's use those words, so you get used to them:
Example: Alex wants to see how many times a "double" comes up when throwing 2 dice.
Each time Alex throws the 2 dice is an Experiment.
It is an Experiment because the result is uncertain.
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}
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}
These are Alex's Results:
| Experiment | Is it a Double? |
| {3,4} | No |
| {5,1} | No |
| {2,2} | Yes |
| {6,3} | No |
| ... | ... |
After 100
Experiments, Alex has 19 "double"
Events ... is that close to what you would expect?
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