An experimental ice cream machine serves one dessert at a time. Each d

Are the events independent or dependent? If it is independent, then the maximum possible probability should be 0.6* 0.8*0.7=0.33
But if dependent, then it should be 0.6. I'm not sure about the solution. It would be great if you could provide us with it.

I found the following formula online to calculate the min and max probability of two mutually non-exclusive event (meaning they can occur at the same time):
Minimum P(A and B): max(0, P(A) + P(B) - 1)
Maximum P(A and B): min(P(A), P(B))

Thus min P(C and xS and B) will be: ((0.6+0.7 - 1) +0.8 -1) = 0.1 i.e. 10%
And max P(C and xS and B) will be min(0.6, 0.7, 0.8) = 0.6 i.e. 60%


Please confirm if this correct or I'm missing still something here?

Initially, I assumed events to be independent. I thought there should be one value of probability, but lately, I realised that nowhere in the question is it defined that events are independent, and that's how min and max came into the picture.

What could be the max probability?
The highest chance Maya gets what she wants is limited by the smallest of the three individual chances. That is:
MaxP =min(60%,80%,70%) = 60%
MaxP = min(0.6,0.8,0.7)= 0.6
Because even if the other two are always true, if only 60% come in cones, it can’t be more than that.

Now about minP
MinP = max(0, 60%+80%+70%−200%) = max(0, 210%−200%) = 10%
MinP = max( 0, 0.6+0.8+0.7−2 ) = max(0,2.1−2) = 0.1 .

Underlying concept - Frechet's bounds
For two events A, B:
max[0,P(A)+P(B)-1] ≤P(A∩B) ≤ min[P(A),P(B)]

Similarly for three events A, B:
max[0,P(A)+P(B)+P(C)-2] ≤ P(A∩B∩C) ≤ min[P(A),P(B), P(C)]

1. Note that each of the features in the ice cream machine isn't stated to be independent from one another. Since they are somewhat dependent, we can imagine them "overlapping" each other and the probability being the intersection.

2. Let's first figure out the probability of getting each feature in the ice cream that Maya wants:

- Cone. "There's a 60% chance it comes in a cone; otherwise, a bowl", so it's 60%.
- Blueberry. "There's an 80% chance it has blueberry; otherwise, no blueberry", so it's 80%.
- Strawberry. "There's a 30% chance it has strawberry; otherwise, no strawberry", so it's 100% - 30% = 70%.

3. The 3 features are 60%, 80%, and 70%. The maximum probability is the greatest intersection we can get with those groups, which is 60%. On the other hand, the minimum probability is the least intersection we can get, while being in the 100% area, which is 10%.

4. Our answer will be: Max - 60% and Min - 10%.