A community center has 30 volunteer tutors. The table summarizes the tutors by prior tutoring experience and whether they can conduct a tutoring session in Spanish.
| Prior tutoring experience | Can conduct a session in Spanish | Cannot conduct a session in Spanish |
|---|---|---|
| Has tutored before | 8 | 10 |
| Has not tutored before | 3 | 9 |
Two different volunteer tutors are selected at random, without replacement. Given that exactly one of the selected tutors can conduct a tutoring session in Spanish, what is the probability that the two selected tutors have different prior tutoring experience?
For a conditional probability involving two selected people, restrict attention to outcomes that satisfy the condition before counting anything else. Here, the condition forces each pair to contain one tutor from each Spanish-language category. Count all such pairs with multiplication, count the favorable pairs by separating the possible experience-level arrangements, and take favorable divided by total.
Hints
Use the given condition first
Because exactly one selected tutor can conduct a session in Spanish, each possible pair has one tutor from the Spanish-speaking column and one tutor from the other column.
Separate the favorable cases
A pair has different experience levels in two ways: the Spanish-speaking tutor has prior experience, or the Spanish-speaking tutor does not have prior experience.
Use a ratio of counts
Count the qualifying pairs with different experience levels, then divide by the total number of pairs with exactly one Spanish-speaking tutor.
Desmos Guide
Set up the counts first
Direct counting is faster than graphing for this question. Identify the total number of qualifying pairs as one Spanish-speaking tutor times one non-Spanish-speaking tutor.
Verify the fraction with calculations
In Desmos, enter (8*9+3*10)/((8+3)*(10+9)). The numerator represents the two favorable pair types, and the denominator represents all pairs with exactly one Spanish-speaking tutor.
Keep the exact fraction
Use the numerator and denominator from the calculation to write the probability as a fraction in lowest terms rather than using only the displayed decimal.
Step-by-step Explanation
Count all pairs that meet the condition
There are tutors who can conduct a session in Spanish and who cannot.
A pair with exactly one Spanish-speaking tutor can be formed in ways.
Count the favorable pairs
For the tutors to have different prior tutoring experience, there are two possible pair types:
-
A tutor who has tutored before and can conduct a session in Spanish paired with a tutor who has not tutored before and cannot: pairs.
-
A tutor who has not tutored before and can conduct a session in Spanish paired with a tutor who has tutored before and cannot: pairs.
Therefore, the number of favorable pairs is
Form the conditional probability
Divide the favorable pairs by all pairs that satisfy the given condition:
The numerator and denominator have no common factor, so this fraction is already in lowest terms.