Dependent events
Two events are independent if the outcome of each event does not affect the outcome of the other event. The two events are not influence by each other. Two events that are not independent are called dependent.
If two events are independent then the following formula will be true: P(A \cap B) = P(A) \times P(B). Given data from an experiment we can check to see if this is significantly close.
Examples
Example 1
A standard six-sided die is rolled 691 times.
If it lands on a six 108 times, what is the probability that the next roll will land on a six?
Is the outcome of the next roll independent of or dependent on the outcomes of previous rolls?
Two events are independent if the outcome of the each event does not affect the outcome of the other event. Two events that are not independent are called dependent.
To check if two events are independent you can use: P(A \cap B) = P(A) \times P(B)
Replacement
When selecting an card from a deck of cards, if we want to select a second card we have two choices:
Putting the card back in the deck before selecting another card
Keeping the card and selecting another card
The first method is described as "with replacement" because the card is placed back into the deck. The second method is called "without replacement" because the card is not placed back into the deck.
When selecting objects from a group "with replacement" selections are independent. Each time you select a marble or a card, you have the same probabilities each time. For example, selecting a red card will be \dfrac{26}{52}=\dfrac{1}{2} every time if you replace the card.
When selecting objects from a group "without replacement" selections are dependent. Each time you select a marble or a card, you change the probabilities for the next selection. For example, selecting a red card will be \dfrac{26}{52}=\dfrac{1}{2} the first time, however if you selected a red card on the first go, then the probability of selecting a red card on the second go is now \dfrac{25}{51}. There was one less red card to choose from and overall only 51 cards to pick from.
Examples
Example 2
A number game uses a basket with 10 balls, all labelled with numbers from 1 to 10. One ball is drawn at random, then another ball is drawn at random.
What is the probability that the ball labelled 2 is picked once if the balls are drawn with replacement?
What is the probability that the ball labelled 2 is picked once if the balls are drawn without replacement?
Events that occur "with replacement" have the item placed back into the group before each selection. Each selection is independent of the others.
Events that occur "without replacement" have the item remain outside of the group after selection. Each selection is dependent of the others. The probabilities will change each selection depending on previous selections.