Question

the spinner is divided into 4 equal parts: yellow (y), green (g), blue (b), and red (r). dorian says that if you spin the spinner twice, then there are exactly 12 possible outcomes. he lists the outcomes in a table. which statement explains dorian’s mistake? spinner image: divided into four equal parts: red, yellow, blue, green table: spin 1: y, g, b, r; spin 2: b g r, y b r, y g r, y g b options: 1. the spinner can land on the same color twice in a row, so there should be 16 possible outcomes. 2. there are 4 possible colors for each spin, so there should be 8 possible outcomes. 3. there are 6 combinations that are counted twice, so there should be half as many possible outcomes. 4. the spinner has 4 colors, so there should be 4 possible outcomes.

Explanation:

Step1: Analyze the spinner spins

Each spin has 4 possible outcomes (Y, G, B, R). For two spins, the total number of possible outcomes is calculated by the multiplication principle: \( 4\times4 = 16 \). This is because for each outcome of the first spin, there are 4 outcomes for the second spin. Dorian's mistake was not considering that the spinner can land on the same color twice (e.g., YY, GG, BB, RR are also valid outcomes which he didn't include), leading him to an incorrect count of 12 instead of 16.

Step2: Evaluate the options

• Option 1: "The spinner can land on the same color twice in a row, so there should be 16 possible outcomes." This correctly identifies that the same - color - twice outcomes were missed, and the total should be \( 4\times4 = 16 \).
• Option 2: "There are 4 possible colors for each spin, so there should be 8 possible outcomes." The multiplication principle for independent events (each spin is independent) is not addition but multiplication. So this is wrong.
• Option 3: "There are 6 combinations that are counted twice, so there should be half as many possible outcomes." There is no over - counting of combinations in the way described here. The main issue is under - counting by not including same - color - twice outcomes. So this is wrong.
• Option 4: "The spinner has 4 colors, so there should be 4 possible outcomes." This ignores the fact that we are spinning twice, and each spin contributes to the outcome. So this is wrong.

Answer:

The spinner can land on the same color twice in a row, so there should be 16 possible outcomes.