Calculate probability coffee will remain in both machines, by the end of the day

Problem description:
In the mall, two identical machines sell coffee.
The probability of the event the machine runs out of coffee by the end of the day is 0.4.
The probability coffee ends in both machines is 0.22.
What is the probability coffee will remain in both machines, by the end of the day.

Problem solution #1:
Let's define event A as coffee ends in the first machine, event B as coffee ends in the second coffee-machine.
As P(A) × P(B) = 0.4 × 0.4 = 0.16 ≠ 0.22, then event A and event B are dependent events.
So P(A∪B) - probability that the coffee will end at least in one of the machines, according to the formula for adding probabilities is:
P(A∪B) = P(A) + P(B) - P(A∩B) = 0.4 + 0.4 - 0.22 = 0.58
Therefore, the probability of the opposite event (coffee will remain in both machines) is:
1 - P(A∪B) = 1 - 0.58 = 0.42

Answer: coffee will remain in both machines, by the end of the day with probability 0.42.


Problem solution #2:
The probability that by the end of the day the coffee will remain in the machine is equal:
1 - 0.4 = 0.6
The probability that coffee will remain in at least one machine is equal:
1 - 0.22 = 0.78
Let P be the probability of remaining coffee in both machines.
Then, using the formula for adding probabilities, we obtain the equation:
0.6 + 0.6 - P = 0.78
Solving this equation, we find
P = 0.42

Answer: the probability of remaining coffee in both machines is 0.42.


Next probability problem solution: Find the probability three balls randomly taken out of the box will be red

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Tuesday, August 11, 2020
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