probability

By example, part #1 (hypothetical probabilities)

Let us say in a covid-19 scenario that four people of a bubble which were not infected before are affraid if one of them is infected after a exposition that two of them had going to the supermarket 5 days ago.

Let's say that due to exposition and taking into account some precautions they had the probability of one of them being infected in that event is 30%, i.e., P(A) = 0.3 and P(B) = 0.3 (probability of one person or another being infected). So, the probabliity of one of them being infected is:

P(A∪B) = P(A) + P(B) - P(A∩B) = 0.3 + 0.3 - 0.09 = 0.51

It is the probability of A or B or both occurring. It is actually calculated by P(A) + P(B) minus the probability of A and B both occuring (i.e. P(A∩B)). The probability of both occurring is given by P(A∩B) = P(A) × P(B), which is this example is 0.3 × 0.3 = 0.09.

Culculate at calculator.net

Example #2 (hypothetical probabilities)

Let us say now the family has considered 51% a high risk so they did a strict isolation these 5 days. Since they live in a bubble we are assuming that one of the four members are likely to be infected from that event on. However, it is expected that some symptoms rise even in mild conditions.

Let's assume that the probability of one present symptoms in five days is 30% (P(A) = 0.3), and in seven days it is 60% (P(B) = 0.6). In this case we have to consider that A can happen 4 times as well as B. The probability of A occurring one time of four is given by 1 - (1 - P(A))^4, so:

Probability of A occuring one in four = 1 - (1 - 0.3)^4 = 0.7599 Probability of B occuring one in four = 1 - (1 - 0.6)^4 = 0.9744

It means that in the current date, 5 days after the exposition, the probability of at least one of the four to present some symptom is about 76%. If they wait 2 more days, the probability of noticing a symptom if they are infected is 97%.

Calculated at calculator.net