Sample Space Confusion for sample space for spinner w/ 2 red parts, 2 blue parts

Julton

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I am working through probability questions in a maths book.

The Sample Space is defined as the "list of all possible outcomes".

The probability of an event is defined as P(E) = Number of favourable outcomes/Number of Possible outcomes.

I am OK with the formula above.

There is a question with a circular spinner, divided into four quadrants: two are coloured red and two are coloured blue. It says to state the Sample Space. I put down {Red, Red, Blue, Blue} the answer at the back of the book is {Red, Blue}. It then asks to calcuate the probability of the spinner landing on red. I calculated P(Red) = Number of favourable outcomes/Number of possible outcomes = 2/4 = 1/2 same answer but different sample spaces.

Another question is a spinner divided into five segments, 1 green, 2 yellow and 2 red. Again to list the sample space I put down {Green, Yellow, Yellow, Red, Red} they list {Green, Yellow, Red}. To find the probability of the spinner landing on green I put: P(Green) = Number of favourable outcomes/Number of possible outcomes = 1/5. The book has the same answer but I can't see how they worked it from the probability space with only 3 outcomes.

I have been on lots of websites, some list sample spaces similar to mine - each colour is listed as many times as it appears and the calculation follows the formula. Most however, seem to follow the book - each colour is listed once. They get the same answers but don't use the formula, they just seem to count each colour as a different outcome when they do the calculation.

Am I missing a trick or is it just up to the person how they wish to do the sample space.

Any insight would be appreciated
 
I am working through probability questions in a maths book.

The Sample Space is defined as the "list of all possible outcomes".

The probability of an event is defined as P(E) = Number of favourable outcomes/Number of Possible outcomes.

I am OK with the formula above.

There is a question with a circular spinner, divided into four quadrants: two are coloured red and two are coloured blue. It says to state the Sample Space. I put down {Red, Red, Blue, Blue} the answer at the back of the book is {Red, Blue}. It then asks to calcuate the probability of the spinner landing on red. I calculated P(Red) = Number of favourable outcomes/Number of possible outcomes = 2/4 = 1/2 same answer but different sample spaces.

Another question is a spinner divided into five segments, 1 green, 2 yellow and 2 red. Again to list the sample space I put down {Green, Yellow, Yellow, Red, Red} they list {Green, Yellow, Red}. To find the probability of the spinner landing on green I put: P(Green) = Number of favourable outcomes/Number of possible outcomes = 1/5. The book has the same answer but I can't see how they worked it from the probability space with only 3 outcomes.

I have been on lots of websites, some list sample spaces similar to mine - each colour is listed as many times as it appears and the calculation follows the formula. Most however, seem to follow the book - each colour is listed once. They get the same answers but don't use the formula, they just seem to count each colour as a different outcome when they do the calculation.

Am I missing a trick or is it just up to the person how they wish to do the sample space.

Any insight would be appreciated

In order to use this definition of probability, it is necessary that each of the "possible outcomes" have the same probability -- they must be "equally likely outcomes". So the sample space they list, while valid in itself, is incompatible with their definition of probability. The sites that do as you do are more consistent. (There are other ways to define probability that do not require a sample space of equally likely outcomes.)

If the sample space does not consist of equally likely outcomes, then you either have to break it up as you did to do the calculation, or replace the number of outcomes with the sum of their probabilities.

Does the book have any worked examples of this sort? Can you show us their work, so we can see what method they are teaching?
 
I am working through probability questions in a maths book. The Sample Space is defined as the "list of all possible outcomes".
The probability of an event is defined as P(E) = Number of favourable outcomes/Number of Possible outcomes. I am OK with the formula above.

There is a question with a circular spinner, divided into four quadrants: two are coloured red and two are coloured blue. It says to state the Sample Space. I put down {Red, Red, Blue, Blue} the answer at the back of the book is {Red, Blue}. It then asks to calcuate the probability of the spinner landing on red. I calculated P(Red) = Number of favourable outcomes/Number of possible outcomes = 2/4 = 1/2 same answer but different sample spaces.

Another question is a spinner divided into five segments, 1 green, 2 yellow and 2 red. Again to list the sample space I put down {Green, Yellow, Yellow, Red, Red} they list {Green, Yellow, Red}. To find the probability of the spinner landing on green I put: P(Green) = Number of favourable outcomes/Number of possible outcomes = 1/5. The book has the same answer but I can't see how they worked it from the probability space with only 3 outcomes.

I have been on lots of websites, some list sample spaces similar to mine - each colour is listed as many times as it appears and the calculation follows the formula. Most however, seem to follow the book - each colour is listed once. They get the same answers but don't use the formula, they just seem to count each colour as a different outcome when they do the calculation.

Am I missing a trick or is it just up to the person how they wish to do the sample space.

Any insight would be appreciated
It think that you need to put your question to your teacher. Almost nothing in mathematics like this has a standard answer.
In my classes I would say for the second example there are three outcomes with different possible frequencies.

Here is example in which it is really to say which is the better way. Toss three dice. Add up the three faces to get the outcome. Now there are sixteen possible outcomes \(\displaystyle 3\text{ to }18\). But the sample space contains \(\displaystyle 6^3=216\) elementary events (it is a mess to list).
Look at the following HERE.
Look at the term \(\displaystyle 21x^{13}\) that tells us that the outcome sum of \(\displaystyle 13\) occurs \(\displaystyle 21\) times. So \(\displaystyle \mathcal{P}(S=13)=\dfrac{21}{216}\).
 
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