Welcome to conditional probability! Conditional probability is the probability of an event occurring given that another event has already occurred. The formula is P of A given B equals P of A intersect B divided by P of B, where P of B must be greater than zero. This diagram shows two overlapping events A and B, where the intersection represents both events occurring together.
Let's work through a practical example using a standard deck of cards. A deck has fifty-two cards: twenty-six red cards consisting of thirteen hearts and thirteen diamonds, and twenty-six black cards. Our question is: what is the probability that a card is a heart, given that it is red? Let A be the event that a card is a heart, and B be the event that a card is red. Using our formula, P of A intersect B equals thirteen over fifty-two, since there are thirteen hearts. P of B equals twenty-six over fifty-two, since there are twenty-six red cards. Therefore, P of A given B equals thirteen over fifty-two divided by twenty-six over fifty-two, which simplifies to thirteen over twenty-six, or one-half.
Another effective method for solving conditional probability problems is using tree diagrams. The steps are simple: first, draw branches for the given condition. Second, add probabilities on each branch. Third, calculate the conditional probability. For our card example, given that the card is red, we have two branches: heart or diamond. Each has a probability of thirteen over twenty-six. So the probability of getting a heart given that the card is red is thirteen over twenty-six, which equals one-half. Tree diagrams make it easy to visualize conditional probability problems.
Let's examine a real-world application with a medical test example. A disease affects one percent of the population. The test is ninety-five percent accurate for sick people and ninety percent accurate for healthy people. The question is: if someone tests positive, what is the probability they actually have the disease? Using Bayes' theorem, we define D as having the disease and T-plus as testing positive. The formula is P of D given T-plus equals P of T-plus given D times P of D, divided by P of T-plus. We calculate P of T-plus as zero point nine five times zero point zero one plus zero point one times zero point nine nine, which equals zero point one zero eight five. Therefore, P of D given T-plus equals zero point nine five times zero point zero one divided by zero point one zero eight five, which is approximately zero point zero eight eight. Surprisingly, there's only an eight point eight percent chance of actually having the disease!
To summarize what we've learned about conditional probability: First, conditional probability measures the likelihood of an event given that another event has occurred. Second, we use the formula P of A given B equals P of A intersect B divided by P of B. Third, tree diagrams provide a visual method for solving these problems. Fourth, Bayes' theorem extends conditional probability to real-world applications like medical testing. Finally, understanding conditional probability helps us avoid common misconceptions in probability reasoning and make better decisions when dealing with uncertain information.