# 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…
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Instructor
Jennifer Beddoe

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

Expert Contributor
Robert Ferdinand

Robert Ferdinand has taught university-level mathematics, statistics and computer science from freshmen to senior level. Robert has a PhD in Applied Mathematics.

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.

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## Practice Questions (please show all work)

1. A box has 6 red marbles, 7 blue marbles and 8 green marbles. You draw one marble at random from the box. What is the probability that the marble is red or green?

2. A die has sides numbered 1-9. You roll the die. What is the probability that the number you obtain is odd or prime.

3. A roulette wheel has slots numbered 1-38 and the number 0. What is the probability that the wheel lands on a number divisible by 4 when you roll the roulette?

1. Let A be the event that the marble is red and B be the event that the marble is green.

Since A and B are not overlapping, that marble cannot be both red AND green. So we get the probability as follows:

• P(A or B) = P(A) + P(B)
• P(A or B) = 6 / (6 + 7 + 8) + 8 / (6 + 7 + 8)
• P(A or B) = 6 / 21 + 8 / 21
• P(A or B) = 14 / 21
• P(A or B) = 2/3

2. Let A be the event that the number is odd and B the event that the number is a prime.

Then A and B are overlapping events since a number can be both prime and odd from among the numbers 1-9.

The odd numbers in 1-9 are 1, 3, 5, 7 and 9 (total of five), while the prime numbers in 1-9 are 1, 3, 5 and 7 (total of four). Hence the numbers in 1-9 that are odd AND prime are 1, 3, 5 and 7 (total of four).

Hence:

• P(A or B) = P(A) + P(B) - P(A and B)
• P(A or B) = 5 / 9 + 4 / 9 - 4 / 9
• P(A or B) = 5/9

3. Let A be the event that the number is divisible by 4. Such numbers are 0, 4, 8, ... , 36 for a total of ten such numbers. The total number of possible outcomes of the experiment are 39.

Hence:

P(A) = 10/39.

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