011 - Probability After Sense Question

Robot Measurement and Belief Update

Initial Setup

• Measurement environment consists of five cells: x1, x2, x3, x4, x5.

• Cell Colors:

  • Red Cells: x2, x3

  • Green Cells: x1, x4, x5

• Uniform Probability for each cell: 0.2

Sensing Measurement

• Robot senses a red color.

• This measurement updates the beliefs about the likelihood of being in each cell.

Update Rule

• Any cell with the correct color (red for x2 and x3): multiplied by 0.6.

• Any cell with green color (x1, x4, x5): multiplied by 0.2.

Calculation of New Beliefs

• For x1 (Green): 0.2 * 0.2 = 0.04

• For x2 (Red): 0.2 * 0.6 = 0.12

• For x3 (Red): 0.2 * 0.6 = 0.12

• For x4 (Green): 0.2 * 0.2 = 0.04

• For x5 (Green): 0.2 * 0.2 = 0.04

Summary of Results

• After applying updates based on measurement:

  • x1: 0.04

  • x2: 0.12

  • x3: 0.12

  • x4: 0.04

  • x5: 0.04

Conclusion

• The probabilities reflect that the belief in being in x2 and x3 (red cells) has increased due to the measurement, while beliefs in x1, x4, and x5 (green cells) have decreased.