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Either/Or Probability: Overlapping and Non-Overlapping Events

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  • 0:06 What Are the Chances?
  • 0:57 Either/Or Probability
  • 3:52 Either/Or Probability…
  • 6:23 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe
Statistics is the study and interpretation of a set of data. One area of statistics is the study of probability. This lesson will describe how to determine the either/or probability of overlapping and non-overlapping events.

What Are the Chances?

'What are the chances?' It's probably a question you hear or say quite often, and it can be used in many different situations:

'What are the chances of winning the lottery tonight?'

'What are the chances of our team winning today?'

'What are the chances you'll get an A on the next test?'

Determining the chances of an event occurring is called probability. Probability is most often written as a percent, but it can also be written as a fraction. The higher the percent, or the closer the fraction is to one, the greater the likelihood that an event will occur. If you have a 90% chance of passing your test, it is quite likely you will pass. On the other hand, if you have a 1/1,000,000 chance of winning the lottery, you are better off saving the cost of the ticket.

Either/Or Probability

Either/or probability refers to the probability that one event or the other will occur. For example, what is the probability that you will draw a Jack or a three from a normal deck of cards? Or, what is the probability that you will roll a 3 or a 5 when rolling a normal 6-sided die? To solve this type of probability problem, here is the formula you will use:

P(A or B) = P(A) + P(B)

To find the probability of each event, simply divide the amount of favorable events by the amount of total events. A favorable event is an event that you want to occur. In the earlier card question, the favorable event is drawing either a Jack or a three. The total number of events is the total number of things that could occur, whether favorable or not.

So, to continue on and solve this card drawing question, we have determined that A is the probability of drawing a Jack, and B is the probability of drawing a three.

There are 4 Jacks in a normal deck of cards, so the number of favorable events (drawing a Jack) is 4. The total number of events is 52 since there are 52 cards in a deck of cards. This means that the probability of drawing a Jack is 4/52, which can be reduced to 1/13.

P(B), or the probability of drawing a three, is also 1/13 because there are 4 threes in a deck of cards and, as before, there are 52 total cards in the deck.

To finish answering the question and find the probability of drawing either a Jack or a three, we use the equation P(A or B) = P(A) + P(B). P(A or B) is equal to 1/13 + 1/13, which is 2/13

To solve the dice question mentioned earlier, follow the same steps. P(A), or the probability of rolling a 3, is 1/6. There is one 3 (the favorable event) and 6 sides on the die (the total events).

P(B) is the probability of rolling a 5 and it's the same, 1/6. Therefore, the probability of rolling either a 3 or a 5 is P(A or B) is equal to 1/6 + 1/6, which is 2/6, or 1/3.

These events are called non-overlapping events, or events that are independent of each other. There are also overlapping events, which are events that are not independent of each other.

Either/Or Probability of Overlapping Events

An example of an overlapping event would be, 'What is the probability of drawing a seven or a diamond from a standard deck of cards?' Since there is a seven of diamonds, there is one card in the deck that is both a seven and a diamond. This has to be accounted for in the equation. If you don't, you will end up with the incorrect probability. The equation for determining the either/or probability of overlapping events is:

P(A or B) = P(A) + P(B) - P(A and B).

As you can see, you must subtract out the probability of the overlapping event to get the right answer. The first event (drawing a seven), has a probability of 4/52 because there are 4 sevens in the deck. The second favorable event (drawing a diamond) is 13/52. The overlap event (drawing the seven of diamonds) has a probability of 1/52.

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