Question

There are many definitions of probability. The main one is that it is a measure of the likelihood of occurrence of events and has a value between 0 and 1. The closer the value is to 1, the more likely the event is....So, an event with probability 0 that mean it will never occur.

In a random experiment, the set of all possible outcomes are defined as the sample space and sometimes can be listed and sometimes can't. Sometimes our interest is limited to certain outcomes that make a certain event. For example, if you roll 2 dice, there are 36 possible outcomes in total. But, you may be interested in the event of getting a sum of 10, rather than all 36 possible outcomes that make up the sample space. In this case, if you denote the event by A, then A= the event of getting a sum of 10 consists of three outcome, in particular A = { (4,6), (5,5), (6,4)}. Assuming that the dice are fair (not biased), then P(A) = The probability of the event A =#(A)/#(S) = 3/ 36 and so on. If you let the event B be at least one of the dice is 4,in this case   B ={ (1,4), (2,4), (3,4),(4,4), (5,4) ,(6,4), (4,1),(4,2), (4, 3),(4,5), (4,6)},

P(B)= 11/36, P(A and B) = 2/36, and P(A union B) =12/36. So, Find P(A given B)= P(A|B) = P(A and B)/ P(B) =(2/36)/ (11/36) =2/11, which is more than twice P(A). Right?

Obviously, A and B are not mutually exclusive but Are they independent? (Hint: Just look at the example and the definition of independence)

In general, if you have 2 events A and B, what does it mean for these 2 events to be independent? How about mutually exclusive? Can they be both? Explain.

Answer 1

P(A|B) came out as 2/11 = 0.1818

P(A) = 3/36 = 0.08333

twice P(A) = 0.16666

So P(A|B) is greater than twice P(A)

If P(AB) = P(A) * P(B) then the two sets A & B are said to be independent

Here P(AB ) = 2/36

P(A) = 3/36

P(B) = 11/36

P(A) * P(B) = (3/36)*(11/36)

= 33 / 1296

= 0.0254

P(AB) = 0.05555

P(AB) is not equal to P(A) * P(B)

So A and B are not independent events

Two events A and B are said to be independent when if the occurrence of one of A or B is not affected by occurrence of the other

If two events A and B are independent then the conditional probability of A given B P(A|B) = P(A)

f two events A and B are independent then the conditional probability of B given A P(A|B) = P(B)

Two events A and B are said to be mutually exclusive if they both don't occur at the same time. The probability of A and B happening or occurring at the same time is 0

If A and B are mutually exclusive, then A and B cannot be independent

If A and B are independent, then A and B cannot be mutually exclusive

For two events A and B to be independent P(AB) = P(A) * P(B)

For two events A and B to be mutually exclusive P(AB) = 0

So if they have to be both then P(A) * P(B) should be 0

But for events with non-zero probability P(A) * P(B) will never be equal to 0

so they can't be independent and mutually exclusive.