Example 1 A spinner has three different colours red (x3), blue (x2) and green (x2). Find:  a) P (Red & Blue)  b) P (At Least 1 Blue) a) We need to find ways to get red & blue. There are two possibilities either Red (1) & Blue (2) or Blue (1) & Red (2). To find the probability of these combined events, we need to multiply the probabilities of red and blue and blue and red. Using the tree diagram we have found the combined probability (working out in the red boxes). Since both combined events are the same (just the other way around), the answers are identical. Now we simply need to add together the two results to see how likely both of these possibilities are to happen. The solution is P (red & blue) = 12/49 b) We need to find all the ways to get at least one blue. These are: RB BR BB BG GB We need to find the probability of all these combined events. The working out is in the blue boxes. Now we simply need to add all these combined probabilities together.
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Probability of Single or Combined Events

Probability of Single Events

Independent events appear quite frequently in our everyday life. Let’s learn how to calculate their probability. Their is a very simple formula to calculate the probability of an event: Let’s look at an example. We have a bag with 9 marbles (3 red, 4 yellow, 2 green) and we want to find the probability that we will get a red marble when taking one out of the bag. The number of appropriate outcomes is 3 as there are three red marbles. The total number of outcomes is 9 as there are nine marbles in the bag. Consequently, the probability is 3/9 or 1/3. We can also write probability as percentage or decimals: 33.333...% or 0.333....

Solving The Probability of Single Events using Tree Diagrams

We can also use tree diagrams to represent probability! Again we can use either fractions decimals and percentage. Mostly we do not use tree diagrams for single event probability. It is only when finding the probability of multiple events that they become really useful.

Probability of Combined Events

Finding the probability of combined events is a bit more challenging as we have to multiply probabilities. The easiest way to solve them is to create tree diagrams. Here are a few examples:
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WHERE MATHS IS AT YOUR FINGERTIPS!
Number of appropriate outcomes Number of total outcomes RED YELLOW GREEN
0.333...
0.444...
0.222...
RED YELLOW GREEN RED BLUE GREEN RED BLUE GREEN RED BLUE GREEN RED BLUE GREEN
Spin 1 (1)
Spin 2 (2)
Example 2 We now have a bag with 12 marbles (2 red, 4 blue, 6 green). We have to pick twice (replacing the marble each time) Find:  a) P (Same Colour Twice)  b) P (Not Blue) a) To find the answer to part a we have to look at all the possibilities where we get the same colour twice: RED & RED, BLUE & BLUE and GREEN & GREEN. We then have to calculate the probabilities for these combined events (working out in the red boxes). Lastly, we have to add these probabilities. The solution is P (Same Colour Twice) = 14/36 b) We need to find all the possibilities that do not contain blue. These are: RR RG GR GG We need to find the probability of all these combined events. The working out is in the blue boxes. Now we simply need to add all these combined probabilities together. The solution is P (Not Blue) = 16/36
RED BLUE GREEN RED BLUE GREEN RED BLUE GREEN RED BLUE GREEN
Pick 1 (1)
Pick 2 (2)

Combined Probability without Replacement

You should have now understood how single and combined probability works. However, all of the combined probability questions were with replacement meaning that for example in example 2, we replaced the first marble before picking the second one. You can earn how to solve probability without replacement in the next lesson. Please share this page if you like it or found it helpful!
Probability Without Replacement
21.0 PROBABILITY 21.1 THE LANGUAGE OF PROBABILITY 21.3 PROBABILITY WITHOUT REPLACEMENT
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Chapter 21.2:  Learning Outcomes Students will learn how to calculate the probability of single events! Students will learn how to use and interpret tree diagrams! Students will learn how to calculate the probability of combined events!
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