In probability theory, events are classified as either dependent or independent. These terms are used to describe the relationship between two or more events and their outcomes.
Independent events are those where the occurrence of one event has no effect on the occurrence of the other. The probability of one event occurring does not affect the probability of the other event occurring. For example, when flipping a coin, getting heads on the first flip does not affect the probability of getting heads on the second flip. Each flip is independent of the other.
Dependent events, on the other hand, are those events where the occurrence of one event does affect the probability of the other event occurring. The outcome of one event is dependent on the outcome of the other. For example, drawing two cards from a deck without replacement is a dependent event. The probability of drawing a second card of a certain suit after drawing a card of that suit on the first draw is affected by the removal of that card from the deck.
To illustrate this difference, let’s consider the example of rolling two dice:
- Independent events: Rolling two dice and getting a 4 on the first die and a 3 on the second die are independent events. The probability of rolling a 4 on the first die is 1/6, and the probability of rolling a 3 on the second die is also 1/6. The probability of rolling a 4 on the first die does not affect the probability of rolling a 3 on the second die, so the probability of rolling a 4 on the first die and a 3 on the second die is the product of their individual probabilities, which is 1/36.
- Dependent events: Rolling two dice and getting a sum of 7 is a dependent event. The outcome of the first roll affects the probability of the second roll. If we roll a 1 on the first die, then the probability of rolling a 6 on the second die to get a sum of 7 is 1/6. However, if we roll a 5 on the first die, then the probability of rolling a 2 on the second die to get a sum of 7 is 0, as it is not possible. Therefore, the probability of rolling a sum of 7 is not simply the product of the probabilities of rolling a 1 and a 6 or rolling a 2 and a 5, but rather depends on the possible combinations of rolls that result in a sum of 7.