Probability of two distinct events happening at the same time, or one of them occuring

Question

I'm starting my studies into probability and I'm struggling a bit to comprehend the following problem:

Having:

• Event A, an event that occurs in 2022, that has a chance of 60% of happening
• Event B, a distinct event that occurs in 2023
• If event A does not happen, then B has a change of 10% of occurring

How would you go about to calculate the probability of at least one event happening and of both events happening?

My guts are telling me to see Bayes theorem, but I feel difficulty of using it in this problem.

Thanks in advance for the help.

Answer 1

Bayes theorem is used in this . Let us denote P(A) as probability of event in 2022 , and P(B) of event occuring in 2023 . Now the probability of happening of B when A has not happened is given by

P(B/A') , where A' denotes compliment of A, and
P(B/A')=(P(AUB)-P(A))/P(A') ...(1)

Also P(B/A')= 10% or 0.1 , and P(A)=60% or 0.6 , therefore P(A')= 0.4 . After putting these values in (1) , we get P(AUB)=0.64 , which is the probability of atleast one of the events happening . You can go in similar form to find the other part of the answer .

Answer 2

How would you go about to calculate the probability of at least one event happening and of both events happening?

$\underline{\text{Question 1:}}$

Alternative approach.

Desired probability is $1 - $ probability of neither event happening.

You are given that $p(\neg A) = (0.4)$.

You are given that $p(\neg B | \neg A) = [1 - 0.1] = (0.9).$

Therefore, $p[(\neg A)(\neg B)] = p(\neg A) \times p(\neg B | \neg A) = (0.4) \times (0.9) = (0.36).$

Therefore, the desired probability is $1 - (0.36) = (0.64).$

$\underline{\text{Question 2:}}$

This question can not be solved, because it requires you to know either the overall probability of B happening, or the probability of B happening, given that event A does occur. Both pieces of information are missing from the query. As an illustration of this conclusion, consider the following two contradictory assumptions, each of which is consistent with the information given in the query:

Assumption-1
$p(B|A) = 1.0.$ That is, it is assumed that it is certain that if event A occurs, then event B will also occur.

Then the desired probability of both events occuring is simply $p(A) = 0.6.$

Assumption-2
$p(B|A) = 0.$ That is, it is assumed that it is certain that if event A occurs, then event B will not also occur.

Then the desired probability of both events occuring must be $0$.