A spinner has 3 three sections, each with different point values and areas.
-The 1-point section is 2/3 the area of the spinner.
-The 2 point section is 1/4 the area of the spinner.
-The 3 point section is 1/12 the area of the spinner.
To the nearest tenth of a point, what is the expected value on any one spin?
1(2/3) + 2(1/4) + 3(1/12) =
8/12 + 6/12 + 3/12 = 17/12
Divide 12 into 17.
8/12 3/12 1/12
12/12=1
To find the expected value on any one spin, we need to calculate the sum of the products of the point values and their corresponding probabilities.
Let's denote the areas of the spinner as A1, A2, and A3 for the 1-point, 2-point, and 3-point sections respectively.
Given:
- The 1-point section has an area of 2/3 the total spinner area, so A1 = 2/3.
- The 2-point section has an area of 1/4 the total spinner area, so A2 = 1/4.
- The 3-point section has an area of 1/12 the total spinner area, so A3 = 1/12.
To calculate the probabilities of each section, we divide the area of each section by the total area of the spinner.
Total area of the spinner = A1 + A2 + A3
So, the probability of landing on the 1-point section is P1 = A1 / (A1 + A2 + A3)
= (2/3) / (2/3 + 1/4 + 1/12)
Similarly, the probabilities of landing on the 2-point and 3-point sections are:
P2 = A2 / (A1 + A2 + A3) = (1/4) / (2/3 + 1/4 + 1/12)
P3 = A3 / (A1 + A2 + A3) = (1/12) / (2/3 + 1/4 + 1/12)
Now, let's calculate the expected value:
Expected value = (1-point value) * P1 + (2-point value) * P2 + (3-point value) * P3
= 1 * P1 + 2 * P2 + 3 * P3
Let's calculate the values to get the final result.
To solve this problem, we need to find the probability of landing on each section and multiply it by the point value of that section. The expected value is then the sum of these values.
Step 1: Calculate the probabilities of landing on each section:
The total area of the spinner is divided among the three sections. Let's represent the total area as 1, since it is the entire area of the spinner.
- The 1-point section has an area of 2/3. This means the probability of landing on the 1-point section is (2/3) / 1 = 2/3.
- The 2-point section has an area of 1/4. So, the probability of landing on the 2-point section is (1/4) / 1 = 1/4.
- The 3-point section has an area of 1/12. Hence, the probability of landing on the 3-point section is (1/12) / 1 = 1/12.
Step 2: Calculate the expected value:
Now, let's multiply the probabilities of landing on each section by their respective point values and find the expected value.
- The expected value from the 1-point section = (2/3) * 1 = 2/3.
- The expected value from the 2-point section = (1/4) * 2 = 1/2.
- The expected value from the 3-point section = (1/12) * 3 = 1/4.
Finally, add up the expected values from each section:
Expected value = (2/3) + (1/2) + (1/4) = 8/12 + 6/12 + 3/12 = 17/12.
To the nearest tenth of a point, the expected value on any one spin is approximately 1.4 points.