Probability formulae
In this article we shall explore how to use probability formulae to solve probability problems.
Deriving probability formulae
We can use the Venn diagram to derive the probability formulae
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Let P(A) = a, P(B) = b, P(A n B) = i. We can find the probabilities of only A and the probabilities of only B by subtracting as shown below.
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If the P(S) = a that must mean that the area outside the closed curves has probability
1 – (a + b – i)
That must mean that the probability of A u B is given by;
P(A u B) = (a – i) + (b – i) + i = a + b – i
We have managed to derive the addition rule;
P(A u B) = P(A) + P(B) – P(A u B)
We can rearrange the rule to find the intersection.
P(A n B) = P(A) + P(B) – P(A u B)
You must also know by now that;
P(A’) = 1 – P(A)
We can use both the probability formulae and the Venn diagrams to solve problems.
Solving probability problems
Example
Given that A and B are two events and that P(A) = 0.6, P(B) = 0.7 and P(A u B) = 0.9
Given that A and B are two events and that P(A) = 0.6, P(B) = 0.7 and P(A u B) = 0.9
Find;
- P(A n B)
- P(A’)
- P(A’ u B)
- P(A’ n B)
Answer
- We can use the probability formulae to find P(A n B)
P(A n B) = P(A) + P(B) – P(A u B)
= 0.6 + 0.7 – 0.9
= 0.4 - We can also use the probability formulae here;
P(A’) = 1 – P(A)
= 1 – 0.6 = 0.4 - We know the probability of the intersection. We can use that to draw a Venn diagram to solve the problem. The intersection is 0.4 we subtract to find the rest of the values in the Venn diagram.
[IMAGE] - The P(A’ u B) is the probability ‘not A’ together with B. This means we have to add up all the regions ooutside A.
P(A’ u B) = (0.3 + 0.1) + 0.4 = 0.8
- To find P(A’ n B) we use the formula;
P(A’ n B) = P(A’) + P(B) – P(A’ u B)
We have already found P(A’ u B) therefore;
P(A’ n B) = 0.4 + 0.7 – 0.8 = 0.3
