This problem represents a type of decaying system where a certain fraction of items is removed at each step, akin to a process modeled by exponential decay.
Here's the step-by-step reasoning:
1. Initial Setup: You start with 100 candies, each with an equal probability of landing "M" side up or "M" side down when shaken.
2. First Shake and Remove: When you shake the pan, roughly half of the candies will land with the "M" side up because the probability is 0.5 for each candy. Out of 100 candies, you can expect about 50 candies to be removed on average.
3. Second Shake and Remove: After removing 50 candies, you are left with 50 candies. Again, you shake the pan and about half of those will show "M" side up, leading to approximately 25 candies being removed.
4. Subsequent Shakes and Removes: Repeating this process, each time you shake the pan:
- From 25 remaining candies, about 12-13 will show "M" side up and will be removed.
- From 12-13 remaining candies, about 6-7 will show "M" side up and will be removed.
- From 6-7 remaining candies, about 3 will show "M" side up and will be removed.
- From the remaining 3 candies, about 1-2 will show "M" side up and will be removed.
5. Final Result: After six shakes and removals, very few candies will remain, most probably none.
### Type of Reaction
This model represents an exponential decay process, where at each step, a fixed proportion of the remaining items is removed. Mathematically, if [tex]\( N \)[/tex] is the number of candies at a given step,
[tex]\[ N_{\text{next}} = N \times \frac{1}{2} \][/tex]
since about half of the candies are being removed at each step.
Thus, this is a classic example of exponential decay, where the quantity of candies decreases by a constant factor (1/2) each step.
