Question

Activity 4: Solve each of the problems with a solution. (10 points)
1. Presently, there are 3 leamersin Grade 8.22 of them are boys. If a learneris to be chosen from the class, find
the probability that the learner is: (a) a boy (b) a girl
2. 1. A fair die rolled. Find is the probability of getting:
(a) an even number (b) a '3' (c) a numberless than 4​

Answer 1

Answer:

Example

I roll a fair die. Let A be the event that the outcome is an odd number, i.e., A={1,3,5}. Also let B be the event that the outcome is less than or equal to 3, i.e., B={1,2,3}. What is the probability of A, P(A)? What is the probability of A given B, P(A|B)?

Solution

This is a finite sample space, so

P(A)=|A||S|=|{1,3,5}|6=12.

Now, let's find the conditional probability of A given that B occurred. If we know B has occurred, the outcome must be among {1,2,3}. For A to also happen the outcome must be in A∩B={1,3}. Since all die rolls are equally likely, we argue that P(A|B) must be equal to

P(A|B)=|A∩B||B|=23.

Now let's see how we can generalize the above example. We can rewrite the calculation by dividing the numerator and denominator by |S| in the following way

P(A|B)=|A∩B||B|=|A∩B||S||B||S|=P(A∩B)P(B).

Although the above calculation has been done for a finite sample space with equally likely outcomes, it turns out the resulting formula is quite general and can be applied in any setting. Below, we formally provide the formula and then explain the intuition behind it.

If A and B are two events in a sample space S, then the conditional probability of A given B is defined as

P(A|B)=P(A∩B)P(B), when P(B)>0.

Here is the intuition behind the formula. When we know that B has occurred, every outcome that is outside B should be discarded. Thus, our sample space is reduced to the set B, Figure 1.21. Now the only way that A can happen is when the outcome belongs to the set A∩B. We divide P(A∩B) by P(B), so that the conditional probability of the new sample space becomes 1, i.e., P(B|B)=P(B∩B)P(B)=1.

Note that conditional probability of P(A|B) is undefined when P(B)=0. That is okay because if P(B)=0, it means that the event B never occurs so it does not make sense to talk about the probability of A given B.

Conditional probability

Fig. 1.21 - Venn diagram for conditional probability, P(A|B).

It is important to note that conditional probability itself is a probability measure, so it satisfies probability axioms. In particular,

Axiom 1: For any event A, P(A|B)≥0.

Axiom 2: Conditional probability of B given B is 1, i.e., P(B|B)=1.

Axiom 3: If A1,A2,A3,⋯ are disjoint events, then P(A1∪A2∪A3⋯|B)=P(A1|B)+P(A2|B)+P(A3|B)+⋯.In fact, all rules that we have learned so far can be extended to conditional probability.