Suppose A and B are dependent events. If P(A) = 0.4 and P(B) = 0.3, what is P(A ∩ B)?
P(A ∩ B) = P(A) × P(B)
P(A ∩ B) = 0.4 × 0.3
P(A ∩ B) = 0.12
What is the definition of dependent events?
Dependent events are events where the outcome of one event affects the outcome of another event. For example, if you roll a die and then roll a second die, the outcome of the second roll is dependent on the outcome of the first roll. If you roll a 6 on the first die, then you have a 1/6 chance of rolling a 6 on the second die.
What is the probability of dependent events?
Suppose A and B are dependent events. If P(A) = 0.6 and P(B|A) = 0.5, then what is P(B)?
P(B) = P(A∩B) = P(A)P(B|A) = (0.6)(0.5) = 0.3
What is the formula for dependent events?
The formula for dependent events is:
P(A and B) = P(A) * P(B|A)
This means that the probability of both events happening is equal to the probability of event A multiplied by the conditional probability of event B given that event A has already happened.
What are the conditions for dependent events?
There are two conditions that must be met for events to be dependent:
The first is that the two events must share some kind of relationship. For example, if Event A is “drawing a card from a standard deck of cards” and Event B is “drawing a second card from the same deck without replacement,” then these events are dependent. This is because the second event is impacted by the first event—in other words, the pool of possible outcomes has changed.
The second condition is that the probability of one event must impact the probability of the other event. In our previous example, let’s say that we know nothing about the cards in the deck except that there are 52 of them. This means that the probability of Event A (drawing any one specific card) is 1/52. If we now draw a card and replace it in the deck, the probability of Event B (drawing that specific card again) changes to 2/52—it’s now twice as likely because there are only 51 cards left in the deck. So, in this case, knowing information about one event definitely changes our knowledge about another related event.
How to calculate the probability of dependent events?
There are two types of events-Independent and Dependent. If two events, A and B, are such that the occurrence or non-occurrence of A does not affect the probability of B then we say that A and B are independent events.
For example, tossing a coin and getting a head is an independent event from tossing a coin again and getting a tail. The first toss of the coin does not affect the probability of the second toss.
But if two events A and B are such that the occurrence or non-occurrence of event A affects or changes the probability of event B then we say that A and B are dependent events.
For example, suppose you have a bag containing 5 white balls and 4 black balls. If you draw one ball at random from the bag and do not replace it before drawing again then these two events become dependent as the probability of drawing a black ball changes after the first draw. This is because now there are only 4 black balls in the bag against 5 white balls. So, now your probability of drawing black changes to 4/9 (4 black balls out of 9 total balls).
To calculate the probability of dependent events, we use the following formula:
P(A ∩ B) = P(B|A) × P(A)
What are the applications of dependent events?
There are many everyday situations that involve dependent events. Some examples are: -Choosing a 1st and 2nd place winner in a contest -Picking 2 cards from a deck without replacement -Drawing 2 balls from a jar without replacement
From the above information, we can conclude that if two events are dependent, then the probability of both events happening is equal to the product of the probabilities of each event occurring.