5. The volleyball team has a double-header on Friday. The probability that they will win both games is 38%. The probability that they will win just the first game is 70%, What is the probability that the team will win the second game given that they have already won the first game?

To find the probability that the volleyball team will win the second game given that they have already won the first game, we can use the concept of conditional probability.

Let A be the event that the team wins the first game, and B be the event that the team wins the second game. We are given:

P(A) = Probability that the team wins the first game = 70% or 0.70 P(A and B) = Probability that the team wins both games = 38% or 0.38

We want to find P(B|A), which is the probability that the team wins the second game given that they have already won the first game.

The formula for conditional probability is: P(B|A) = P(A and B) / P(A)

So we have: P(B|A) = 0.38 / 0.70

Now we can compute this value: P(B|A) = 0.38 / 0.70 = 0.542857... (approximately)

Therefore, the probability that the volleyball team will win the second game given that they have already won the first game is about 54.29%.

Extra: Let's delve a little deeper into the concept of conditional probability, which is the probability of an event occurring given that another event has already taken place. The notation P(B|A) reads as "the probability of B given A," showing that we are interested in the likelihood of event B happening once event A is known to have occurred.

An important thing to remember about conditional probability is that it depends on the relationship between the two events. If two events are independent, meaning the occurrence of one event does not affect the occurrence of the other, the conditional probability of one event given the other is simply the probability of the event itself. However, if events are dependent, meaning the occurrence of one does affect the likelihood of the other, the conditional probability must take into account this dependence.

The idea is that with this new information about the first event (A happening), we adjust our probabilities for the second event (B). This adjustment is made by considering how likely both events are to happen together (the conjunction of A and B, written as P(A and B)) compared to how likely the first event is to happen on its own (P(A)).

This concept is widely used in various fields, including statistics, business, medicine, and many areas of science, as it helps us make more informed predictions and decisions when past or present information alters the likelihood of future events