I have two fair spinners. The first has the numbers 1,2,3, and 4. The second has the numbers 2,3,4 and 5. I spin both spinners and add the scores. What is the probability that the total is odd?

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Explanation:

It helps to make a table of all possible outcomes. Refer to the diagram below. Along the top we have the outcomes 1,2,3,4. Along the left side we have 2,3,4,5. Each row & column header is then added to form the corresponding inner cell. For example, in the top left corner we have 1+2 = 3.

There are 4 outcomes of spinner A and 4 outcomes of spinner B. This means we have 4*4 = 16 outcomes overall. Out of those 16 sums, we have the odd results {3,5,7,9} show up 1+3+3+1 = 8 times.

The probability we want is therefore 8/16 = 1/2. This is expected since half of the values of {1,2,3,4} are odd, and the same applies to {2,3,4,5}. So we should expect half of the sums to be odd as well.

Put another way: for any row or column, you'll find that the pattern is odd/even/odd/even or even/odd/even/odd. This shows that 50% of any row or column represents an odd number. This idea extends out to the entire table overall.

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