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You have learned that for independent events, the outcome of one event does not affect the outcome of the other. Classic examples of independent events include tossing a coin and then drawing a card from a deck, or rolling a fair six-sided die and then drawing a marble from a bag. You can use the independent event formulas to find the probability that all the independent events occur, or at least one independent event occurs.
What about dependent events? How do you calculate the probability that both or one of a set of dependent events occurs? In the video below, the instructor will show you how to can use two-way frequency tables to find these types of probabilities. He will also define the term conditional probability. This is the probability that an event occurs, given that another event has already happened. Make sure you understand how conditional probability is related to dependent events as you watch this video.
You may want to follow along using the study guide from earlier in this lesson.
With regards to dependent events, the method becomes more complicated still, since one events’ outcome affects the probability of the other event occurring. The probability that an event, B, will occur given that another event, A, has already occurred is called a conditional probability. Conditional probability exists when two events are dependent. A two-way frequency table, or contingency table, is a table that contains data from two different categories.
It can be used to simplify the process for calculating the probability of conditional, or dependent events. For example, this table shows two categories, the location and education level of employees in a company. What is the probability that a randomly selected person works in Ohio, given that they have earned a high school diploma?
When we say the phrase, “given that” we are implying that we’re only interested in those that earned only a high school diploma, and not any other education level. So for this example we see that there were one thousand one hundred fifty-six employees in Ohio with a high school diploma and a sum total of five thousand six hundred sixty-one employees with a high school diploma.
Therefore the probability is one thousand one hundred fifty-six out of five thousand six hundred sixty-one.
If we reverse the phrasing, we see a different answer.
Another way to think of this is by asking yourself, ‘of those from the state of Ohio, how many have earned a High School diploma’. The previous example might be reworded as, ‘of those that earned a high school diploma, how many are from Ohio’? This should help you see why the probabilities are different. Please try the last example on your own. Pause the video now and resume playback in a moment to check your work
To finish this video, I’d like to share with you the formula needed to compute conditional probability. As long as the probability of the first event occurring is not zero, then the probability the second event occurs, given that the first event occurs is the probability of both events occurring divided by the probability of the first event. I encourage you to try using this formula to check our work for the last three examples. Good Luck!
1. What information is shown in a contingency table?
2. What is meant by the phrase “given that” and how do you indicate this term using mathematical notation?
3. Return to the two-way frequency table shown in the video at time mark 6:53. If you wanted to show the probability that an employee has a college degree given that they are from Indiana, how would you use the P(B |A) notation to express this?
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A contingency table is another name for a two-way frequency table. This kind of display shows data from two different categories. |
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The phrase “given that” means that the first event has already occurred. You can use the “|” symbol to indicate this phrase. |
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In this case, event B is “the employee has a college degree” and event A is “the employee is from Indiana,” so you can express this probability as P(college degree|Indiana). |
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