# Find the probability that both sellers are free

Problem description:
There are two sellers working in a small store.
At a random point in time, each of them can be engaged in customer service with a probability of 0.3.
Moreover, they can be occupied simultaneously with a probability of 0.1.
Find the probability that at a random moment in time
a) both sellers are free;
b) one of them is free, and the other is busy.

Problem solution:
(a)

## Find the probability that at a random moment both sellers are free

Let define event A as seller #1 is busy and event B as seller #2 is busy.
Then P(A∪B) - the probability that at least one of the sellers is busy.

According to the probability addition formula is:
P(A∪B) = P(A) + P(B) - P(A∩B) = 0.3 + 0.3 - 0.1 = 0.5

Then the probability that both sellers are free is:
1 - P(A∪B) = 1 - 0.5 = 0.5.

Answer: probability that both sellers are free is 0.5

(b)

## Find the probability that one seller is busy, and the other is free

In this situation, there are four possible options with sellers:
1) seller #1 is free and seller #2 is busy;
2) seller #2 is free and seller #1 is busy;
3) both sellers are free (probability of this event = 0.5);
4) both sellers are busy (probability of this event = 0.1).
Therefore, the probability of the first and second option is equal to: 1 - 0.5 - 0.1 = 0.4

Answer: probability that one seller is free, and the other is busy equals 0.4

Next probability problem solution: Find the probability that two purchased batteries will be working.

Sunday, October 24, 2021
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