Conditional Probability
Conditional Probability
How to handle Dependent Events
Life is full of random events! You need to get a "feel" for them to be a smart and successful person.
Independent Events
Events can be "Independent", meaning each event is not affected by any other events.
Example: Tossing a coin.
Each toss of a coin is a perfect isolated thing.
What it did in the past will not affect the current toss.
The chance is simply 1-in-2, or 50%, just like ANY toss of the coin.
So each toss is an Independent Event.
Dependent Events
But events can also be "dependent" ... which means they can be affected by previous events ...
Example: Marbles in a Bag
2 blue and 3 red marbles are in a bag.
What are the chances of getting a blue marble?
The chance is 2 in 5
But after taking one out you change the chances!
So the next time: * if you got a red marble before, then the chance of a blue marble next is 2 in 4 * if you got a blue marble before, then the chance of a blue marble next is 1 in 4
See how the chances change each time? Each event depends on what happened in the previous event, and is called dependent.
That is the kind of thing we will be looking at here. "Replacement"
Note: if you had replaced the marbles in the bag each time, then the chances would not have changed and the events would be independent: * With Replacement: the events are Independent (the chances don't change) * Without Replacement: the events are Dependent (the chances change) Tree Diagram
A Tree Diagram: is a wonderful way to picture what is going on, so let's build one for our marbles example.
There is a 2/5 chance of pulling out a Blue marble, and a 3/5 chance for Red:
We can even go one step further and see what happens when we select a second marble:
If a blue marble was selected first there is now a 1/4 chance of getting a blue marble and a 3/4 chance of getting a red marble.
If a red marble was selected first there is now a 2/4 chance of getting a blue
How to handle Dependent Events
Life is full of random events! You need to get a "feel" for them to be a smart and successful person.
Independent Events
Events can be "Independent", meaning each event is not affected by any other events.
Example: Tossing a coin.
Each toss of a coin is a perfect isolated thing.
What it did in the past will not affect the current toss.
The chance is simply 1-in-2, or 50%, just like ANY toss of the coin.
So each toss is an Independent Event.
Dependent Events
But events can also be "dependent" ... which means they can be affected by previous events ...
Example: Marbles in a Bag
2 blue and 3 red marbles are in a bag.
What are the chances of getting a blue marble?
The chance is 2 in 5
But after taking one out you change the chances!
So the next time: * if you got a red marble before, then the chance of a blue marble next is 2 in 4 * if you got a blue marble before, then the chance of a blue marble next is 1 in 4
See how the chances change each time? Each event depends on what happened in the previous event, and is called dependent.
That is the kind of thing we will be looking at here. "Replacement"
Note: if you had replaced the marbles in the bag each time, then the chances would not have changed and the events would be independent: * With Replacement: the events are Independent (the chances don't change) * Without Replacement: the events are Dependent (the chances change) Tree Diagram
A Tree Diagram: is a wonderful way to picture what is going on, so let's build one for our marbles example.
There is a 2/5 chance of pulling out a Blue marble, and a 3/5 chance for Red:
We can even go one step further and see what happens when we select a second marble:
If a blue marble was selected first there is now a 1/4 chance of getting a blue marble and a 3/4 chance of getting a red marble.
If a red marble was selected first there is now a 2/4 chance of getting a blue