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Find and compare the conditional probabilities that a customer is satisfied with their cell phone company.
Company A
We want to choose the best cell phone company to sign with for the next two years. All three plans that we are considering are equally priced. Therefore, we want to choose the company with which clients were most satisfied. We asked several of our neighbors whether or not they are satisfied with their current cell phone company. The given table sums up the results.
| Satisfied | Not Satisfied | |
|---|---|---|
| Company A | |||| | || |
| Company B | |||| | ||| |
| Company C | |||| | | |||| |
Let's translate these tally marks into numbers.
| Satisfied | Not Satisfied | |
|---|---|---|
| Company A | 4 | 2 |
| Company B | 4 | 3 |
| Company C | 6 | 5 |
| Satisfied | Not Satisfied | Totals | |
|---|---|---|---|
| Company A | 4 | 2 | 4+2= 6 |
| Company B | 4 | 3 | 4+3= 7 |
| Company C | 6 | 5 | 6+5= 11 |
| Totals | 4+4+6= 14 | 2+3+5=10 | 6+ 7+ 11= 24 |
The sums of the rows and columns are called marginal frequencies. The total number of observations 24 can be found in the bottom right corner of the table. Now, all of the frequencies in the table can be divided by 24 in order to find the joint relative frequencies and marginal relative frequencies.
| Satisfied | Not Satisfied | Totals | |
|---|---|---|---|
| Company A | 4/24 ≈ 0.167 | 2/24 ≈ 0.083 | 6/24=0.25 |
| Company B | 4/24 ≈ 0.167 | 3/24 = 0.125 | 7/24 ≈ 0.292 |
| Company C | 6/24= 0.25 | 5/24 ≈ 0.208 | 11/24 ≈ 0.458 |
| Totals | 14/24 ≈ 0.583 | 10/24 ≈ 0.417 | 24/24=1 |
We want to find the best cell phone company. We can do this by finding the conditional probability that a customer is satisfied with their company and then comparing these probabilities. Let's denote these probabilities as P_A, P_B, and P_C.
P_A &= P(satisfied | Company A)
P_B &= P(satisfied | Company B)
P_C &= P(satisfied | Company C)
The above probabilities can by found by dividing each joint relative frequency in the Satisfied
column by the marginal relative frequency in its corresponding row. Let's first calculate it for Company A.
P_A = P(satisfied and Company A)/P(Company A)
In the numerator, we will substitute the joint relative frequency 0.167 from the table because it is the probability that a customer of Company A is satisfied. In the denominator, we need to use the marginal relative frequency from the first row 0.25 because it is the probability that a customer uses Company A.
P(satisfied and Company A)= 0.167, P(Company A)= 0.25
Use a calculator
We can calculate probabilities P_B and P_C similarly.
| Formula | Substitution | Probability | |
|---|---|---|---|
| Company A | P_A=P(satisfied and Company A)/P(Company A) | 0.167/0.25 | 0.668 |
| Company B | P_B=P(satisfied and Company B)/P(Company B) | 0.167/0.292 | ≈ 0.572 |
| Company C | P_C=P(satisfied and Company C)/P(Company C) | 0.25/0.458 | ≈ 0.546 |
Finally, we should look for the company which has the greatest probability of its clients being satisfied. Therefore, we should choose Company A.
