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If more than one event is occurring, the events could be dependent or independent.
For independent events, the outcome of one event does not affect the probability of the second event. Since independent events have a unique definition, they also have unique formulas for calculating their probabilities.
In the video below, the instructor will explain the different kind of formulas you can use to calculate the probabilities of independent events. Pay close attention to the notation used.
You may want to follow along using the study guide from earlier in this lesson.
Let us now look at the method for computing the probabilities of independent events. In order to calculate the probability that both independent events A and B occur, we simply multiply the probability of event A occurring times the probability of event B occurring. For example, we said that picking a card, then rolling a single die are independent events, so the probability of picking a King from a deck of cards and rolling a five on a die would be four out of fifty-two times one out of six since there are four kings in a deck of fifty-two cards and only one five on a six sided die. It is common to reduce the resulting fraction into a decimal by dividing the numerator by the denominator.
In the second example of independent events, what is the probability of picking a king, replacing the card, and then picking a red card? In this case, the probability would be four out of fifty-two times twenty-six out of fifty-two.
Please try the last example on your own. Pause the video now and resume playback in a moment to check your work.
Since these events are independent of each other, and there is a three out of six chance of rolling an even number, then three out of six chance of rolling an odd number, the product of these two probabilities is nine out of thirty-six, or 0.25.
The method for computing whether either of two independent events occur, rather than both, is slightly more involved. We first sum the probabilities of A and B, then subtract the probability of A and B. It is more likely that one event OR the other will occur, rather than both. For example, there is a better chance that you pick a King from your deck of cards OR a five, rather than a King AND a five.
You might find it easier to convert each probability to a decimal first, this eliminates the need for adding and subtracting fractions. Please try the last two on your own. You’ll notice that they also use the same premises from before. Pause the video now, and resume playback in a moment to check your work.
How well did you understand the information in the video? Use the activity below to find out. Answer each question based on what you learned from the video instruction.
1. Explain the difference between AND and OR probability.
Which has a higher probability: (1) drawing a jack from a deck of cards and rolling a 2 on a fair die or (2) drawing a jack or rolling a 2 on a fair die? Explain.
Explain how to use the formula \( P\left( A \text{ and } B \right) = P(A) \cdot P(B) \).
| Your Responses | Sample Answers |
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An AND probability tells you the probability that both events occur. An OR probability gives you the probability that at least one of the events occurs. |
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(2) Drawing a jack or rolling a 2 on a fair die will have a higher probability. The probability that one event occurs is always higher than the probability that both events occur. |
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You can use this formula to determine the probability that two independent events occur. You need to find the probability of event A occurring and the probability of event B occurring, and then find the product of their probabilities. |
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