## 2.2 Contigency table and probability

• Another useful way to calculate probabilities is having the data arranged in a contingency table

• Contingency table displays the joint probabilities $$P(A_i \cap B_j)$$ at all possible intersections of row $$i$$ and column $$j$$, the marginal probabilities $$P(A_i)$$ in the summary column, and the marginal probabilities $$P(B_j)$$ in the summary row. The marginal probabilities are consider as unconditional probabilities

• Union probabilities and conditional probabilities can be computed from the joint and marginal probabilities

Example 2.4 Let consider contingency table from Excel data with fifty sampled companies ($$n=50$$) which are grouped according to two qualitative variables: type of branch and company’s listing (if company is listed on the stock exchange or not)
TABLE 2.1: Contigency table (left) and probabilities table (right)
Branch No Yes Total
manufacturing 8 10 18
service 6 5 11
Total 29 21 50
Probabilities $$B_1$$ $$B_2$$ $$P(A_i)$$
$$A_1$$ 0.16 0.2 0.36
$$A_2$$ 0.12 0.1 0.22
$$A_3$$ 0.3 0.12 0.42
$$P(B_j)$$ 0.58 0.42 1
1. What is the probability that a company is listed on the stock exchange?

2. What is the probability that a company deals with trade AND is not listed on the stock exchange?

3. What is the probability that a company deals with trade OR is not listed on the stock exchange?

4. What is the probability that a company deals with trade GIVEN that is not listed on the stock exchange?

5. What is the probability that a company is not listed on the stock exchange GIVEN that deals with trade?

6. Using multiplication rule check if two qualitative variables are independent.

7. Create another contingency table assuming independence between type of branch and company’s listing.