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 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.
Now, if we put all those numbers in the probability equation, we can determine that the probability of drawing either a seven or a diamond from a regular deck of cards is 4/52 + 13/52 - 1/52, which is 16/52, or 4/13. The probability of pulling a seven or a diamond from a normal deck of cards is 4/13.
Let's try one more example:
The numbers 1 through 20 are written on separate slips of paper and placed in a hat. One of the slips is randomly drawn. What is the probability that either an even or a prime number is drawn?
In the numbers from 1 to 20, there are 10 even numbers (2, 4, 6, 8, 10, 12, 14, 16, 18 and 20). Therefore, P(A) is 10/20. There are 8 prime numbers (2, 3, 5, 7, 11, 13, 17 and 19), so P(B) is 8/20.
The overlap between the two groups includes only the number 2. So, P(A + B) is 1/20. The equation is 10/20 + 8/20 - 1/20, which is equal to 17/20.
Probability is the chance that something is or is not going to happen. To determine the probability of an event, you need to determine the number of favorable events and divide that by the number of total events that could happen. To determine the probability of one event or another occurring, you first need to determine if the events are overlapping or non-overlapping. If they do not overlap, then you just need to add the probability of each event occurring together. If there is some overlap, then to get the true probability, you must also subtract out the probability of the overlapping event.
Once you've viewed this video lesson, you might have the knowledge required to:
- Know what it takes to determine an event's probability
- Differentiate between overlapping and non-overlapping events
- Find the probability of both overlapping and non-overlapping events
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?
Answers (to check your work)
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).
- 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.
P(A) = 10/39.