What are independent and dependent events? How about replacement/without replacement?
If
T = the total number of items in a set
A = the total number of items in a subset
B = the total number of items in another subset
Independent events' probabilities are not affected by any other events (if you want to take the probability of two events happening, you would just be multiplying the individual probability of each happening). An example is replacement. If I want the probability of the same event happening with replacement, I take the probability of taking an item out of a set, put it back, and then take the probability again.
Example: p(A,A) = A/T * A/T
However, dependent events' probabilities are affected by other events (taking the probability of it by itself would not be the same as taking the independent events' probabilities). An example is without replacement. If I want the probability of the same event happening without replacement, I take the probability of taking an item out of a set, keep it out, and then take the probability again.
Example: p(A,A) = A/T * (A-1)/(T-1)
ur welcome :)
Independent and dependent events refer to the relationship between two or more events occurring together.
1. Independent Events:
Independent events occur when the outcome of one event does not affect the outcome of another event. In other words, the probability of one event happening does not influence the probability of the other event happening. For example, flipping a coin twice. The outcome of the first flip (heads or tails) does not affect the outcome of the second flip, so the events are independent.
2. Dependent Events:
Dependent events occur when the outcome of one event does affect the outcome of another event. In this case, the probability of one event happening does influence the probability of the other event happening. For example, drawing cards from a deck without replacement. The probability of drawing a certain card changes after each card is drawn since there are fewer cards left in the deck.
3. Replacement:
Replacement refers to the act of returning the item after it has been selected. When replacement is allowed, the probability of the second event remains the same as the first event since the items are returned and the original situation is preserved.
4. Without Replacement:
Without replacement means that the item is not returned after it has been selected. When without replacement occurs, the probability of the second event changes since the chances of selecting different items from the remaining pool have altered.
To understand whether events are independent or dependent, or whether replacement is allowed or not, it's essential to consider the specifics of the situation or problem at hand. These concepts play a crucial role in probability calculations and statistics.