The product rule applies to independent occurrences. Only then is the chance of the intersection equal to the product of the events' probabilities. Only when occurrences A and B are independent does P(AB) equal P(A) + P(B). Otherwise, the chances that they would both occur are not added together.
In this problem, occurrences A and B are dependent on each other. This means that if A occurs, B cannot occur and vice versa. In this case, the chance that they will both occur is included in the answer because they are repeated events.
Since these are repeated events, the chance that they will both occur is already included in the answer. It's 100%. There is no need to calculate it again.
Here's how the problem can be reworded: "If A and B occur with the same frequency, then the chance that they will both occur twice in a row is the same as the chance that they will both occur once. Therefore, the chance that they will occur twice in a row is 25%."
You should know that repeating events increase the likelihood of something happening again. In this case, it's obvious because there are two chances per occurrence that both will happen again. If we were looking at non-repeating events, then the chance that they will both occur twice in a row would be 50%.
Formula P (AB) The P(AB) formula is used to calculate the likelihood of two independent occurrences "A" and "B" occurring concurrently. The sign "" denotes an intersection. The intersection of two or more sets is the set of items that are shared by all sets.
The set of items that are shared by all sets is the intersection of two or more sets. The symbol represents the junction.
P (A) = P (AB) + P (C) is the total probability rule (ABc). BC is the complement of B and implies "intersection." The probabilities required for the computation of total probability aren't always stated in the precise form required to solve the equation. For example, if A and B are mutually exclusive events, then P(A) + P(B) = 1.
Total probability allows us to calculate the probability of an event that cannot be broken down into simple parts. For example, if we know the probability of an accident happening while a driver is texting on his/her phone is 0.05, then we can conclude that the probability of an accident happening while the driver is not texting is 0.95. Total probability enables us to make this conclusion because it requires only two probabilities: the probability of an accident happening and the probability of an accident happening while driving distracted.
Total probability also helps us understand why some accidents happen. For example, if the probability of an accident happening while driving distracted is 0.05 and the probability of an accident happening otherwise is 0.95, then we can conclude that driving distracted is the cause of about half of all accidents. This means that drivers should try to avoid being distracted when they get in their cars!
Finally, total probability can help us understand data from real-life situations.
The likelihood of occurrences A and B occurring simultaneously equals the probability of A and B intersecting. P (A + B) denotes the probability of occurrences A and B intersecting. P (A + B) = 0 if occurrences A and B are mutually exclusive. P (A + B)
In other words, the probability that events A and B will occur together is equal to the probability that they will both occur. This fact can be used to find the probability of any combination of events. For example, the probability that a randomly selected house will not have a dog and not be red is given by p (house - dog - not red). Since dogs are animals and therefore cannot live without water, it is reasonable to assume that they will not be living in this house.
Dogs need to drink at least once every 8 hours. Thus, the probability that a randomly selected house will not have a dog and will not have any water available for him is given by p (house - dog - no water). Since houses with dogs are likely to have at least one room with water, it is reasonable to assume that any house with a dog will also have some kind of water available.
Intersection An intersection is the likelihood that both or all of the events you're calculating will occur at the same moment (less likely). According to the Multiplication Rule, the probability of two occurrences (A and B) is given by: P (A and B) = P (A) x P (B). In other words, the probability that A will happen and then B will happen is just the probability that A will happen times the probability that B will happen if A has happened.
In case of dependent events, it's important to remember that the calculation involves the simultaneous occurrence of both events. If we were to calculate the intersection between two events that weren't dependent, we would simply multiply P (A) by P (B), as these events are assumed to be independent. However, because these events are dependent, we need to take into account how much time has passed since the first event occurred before determining the second event's probability.
Let's say that we want to know the probability of getting a head when we roll a coin twice. We can use the multiplication rule to work out the probability of getting a head on our first try and then another head on our second try: P (getting a head on both tries) = P (getting a head on first try) x P (getting a head on second try | getting a head on first try). So, using the multiplication rule, this probability is equal to 0.5 x 0.5 or 0.25.