# Common Core: High School - Statistics and Probability : Explain Conditional Probability and Independence: CCSS.Math.Content.HSS-CP.A.5

## Example Questions

### Example Question #1 : Explain Conditional Probability And Independence: Ccss.Math.Content.Hss Cp.A.5

A new test has been developed to detect lung cancer. A positive test result indicates lung cancer. There are, however, false positives, where the test incorrectly shows a positive result when there is no disease. Similarly, the test occassionally produces a negative result when a patient does in fact have lung cancer. All of the following are accurate EXCEPT:

Explanation:

### Example Question #2 : Explain Conditional Probability And Independence: Ccss.Math.Content.Hss Cp.A.5

Determine whether or not the following scenario represents a conditional probability.

A doctor wants to determine a patient's risk to heart disease knowing that the patient has a history of high cholesterol.

Cannot be determined

Yes, the probability of heart disease is dependent on high cholesterol levels

Yes, the probability of having high cholesterol is dependent on heart disease

No, the probability of heart disease is independent of cholesterol levels

Yes, the probability of heart disease is dependent on high cholesterol levels

Explanation:

In order to solve this problem, we need to discuss probabilities and more specifically conditional probabilities. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate conditional probabilities. A conditional probability is defined as the probability that event B will occur given the information that an event A has already occurred.

It is expressed using the following equation:

In this equation, the probability of event B given that event A has occurred is equal to the probability of the intersection of events A and B divided by the probability of event A. It is important to note that if the events are independent, then the probability of event B given event A is simply the probability of event B because event A does not affect it.

Now, that we have an understanding of conditional probabilities let's investigate the scenario in the question. In the question, the doctor wants to know a patient's risk of heart disease. This represents the event that may occur. Next, the doctor knows that the patient has a history of high cholesterol. This information represents given information; thus, this scenario represents a conditional probability. This probability may be denoted in the following way:

This is read as "the probability of a patient having heart disease given that he or she has high cholesterol." This relationship is not the reverse because we want to know the risk of heart disease and we know that the patient has high cholesterol; therefore, high cholesterol is not dependent on high cholesterol in this particular scenario. The correct answer is the following: "Yes, the probability of heart disease is dependent on high cholesterol levels."

### Example Question #3 : Explain Conditional Probability And Independence: Ccss.Math.Content.Hss Cp.A.5

Determine whether or not the following scenario represents a conditional probability.

The chances of drawing a jack after drawing a jack of spades and then replacing the card back into the deck.

Yes, the probability of drawing a jack is dependent on the first draw because the card is replaced

No, the probability of drawing a jack is independent of the first draw because the card is replaced

Yes, the probability of drawing a jack is dependent on the first draw because the card is not replaced

Cannot be determined

No, the probability of drawing a jack is independent of the first draw because the card is replaced

Explanation:

In order to solve this problem, we need to discuss probabilities and more specifically conditional probabilities. We will start with discussing probabilities in a general sense. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:

Let's use a simple example: the rolling of a die. We want to know the probability of rolling a one. We know that the sample space is six because there are six sides or outcomes to the die. Also, we know that there is only a single side with a value of one; therefore,

Now, let's convert this into a percentage:

Probabilities expressed in fraction form will have values between zero and one. One indicates that an event will definitely occur, while zero indicates that an event will not occur. Likewise, probabilities expressed as percentages possess values between zero and one hundred percent where probabilities closer to zero are unlikely to occur and those close to one hundred percent are more likely to occur.

Now that we understand the definition of a probability in its most general sense, we can investigate conditional probabilities. A conditional probability is defined as the probability that event B will occur given the information that an event A has already occurred.

It is expressed using the following equation:

In this equation, the probability of event B given that event A has occurred is equal to the probability of the intersection of events A and B divided by the probability of event A. It is important to note that if the events are independent, then the probability of event B given event A is simply the probability of event B because event A does not affect it.

Now, that we have an understanding of conditional probabilities let's investigate the scenario in the question. In the question, we want to know the chances of drawing a jack after drawing a jack of spades and replacing it in the deck. The chances of drawing a jack in the first draw was four out of fifty-two. Now, that the card has been replaced the chances of drawing a jack are four out of fifty-two; however, if the card were not replaced then the chances of drawing another jack would have been reduced to three out of fifty-two. Since, the card was replaced, the probability of drawing a jack on the second draw is independent of the first; therefore, the following choice is correct: "No, the probability of drawing a jack is independent of the first draw because the card is replaced."