Define the concept of independent events with regards to probability. Provide an example of an event that is independent and an event that is not independent
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
I'll let you provide the examples.
The concept of independent events in probability refers to the situation where the occurrence or outcome of one event does not affect the occurrence or outcome of another event. In other words, the probability of one event happening does not depend on whether another event has happened or will happen.
To determine if two events are independent, you can use the formula: P(A and B) = P(A) * P(B), where P(A and B) represents the probability of both events A and B occurring, and P(A) and P(B) represent the probabilities of events A and B occurring individually.
Example of independent events:
Let's say you toss a fair coin twice. The first event can be getting a heads on the first toss, and the second event can be getting a tails on the second toss. Since the outcome of the first toss (heads) does not affect the outcome of the second toss (tails), these events are independent.
To calculate the probability of these independent events, you multiply the probabilities of each event:
P(Heads on 1st toss and Tails on 2nd toss) = P(Heads on 1st toss) * P(Tails on 2nd toss) = (1/2) * (1/2) = 1/4.
Example of dependent events:
Consider drawing two cards from a standard deck without replacement. Let event A be drawing a red card on the first draw, and event B be drawing a red card on the second draw. In this case, the outcome of the first event (drawing a red card) affects the outcome of the second event (as one red card has already been removed from the deck). Therefore, these events are dependent.
To calculate the probability of these dependent events, you multiply the probabilities of each event, taking into account the change in the available outcomes:
P(Red card on 1st draw and Red card on 2nd draw) = P(Red card on 1st draw) * P(Red card on 2nd draw, given Red on 1st draw) = (26/52) * (25/51) = 25/102.