The table summarizes the shift assignments and safety-orientation completion records for volunteers at a community garden. Each volunteer was assigned to exactly one shift.
| Assigned shift | Completed safety orientation | Did not complete safety orientation |
|---|---|---|
| Morning | 48 | 15 |
| Afternoon | 27 | |
| Evening | 72 | 20 |
Of the volunteers who completed the safety orientation, one-third were assigned to the afternoon shift.
If a volunteer is selected at random from the volunteers who were not assigned to the evening shift, what is the probability that the volunteer completed the safety orientation or was assigned to the afternoon shift?
For a conditional-probability table question with a missing entry, first translate any probability statement into a fraction of counts and solve for the missing value. Next, rebuild the sample space using the condition after the word "given" or the restriction in the question. For an "or" event, count one full group and then add only members of the other group who were not already included.
Hints
Use the phrase "of the volunteers who completed"
The denominator in the one-third statement is the total number of volunteers who completed the safety orientation across all three shifts.
Restrict the sample space
The condition that the volunteer was not assigned to the evening shift means that only the morning and afternoon rows belong in the final denominator.
Avoid double-counting
To count volunteers who completed the orientation or were assigned to the afternoon shift, count every afternoon volunteer and then add only morning volunteers who completed the orientation.
Desmos Guide
Use algebra first
Algebra is the fastest method because the missing value comes directly from one conditional-probability equation. Desmos can verify that value and the final fraction.
Verify the missing count
Graph and . Click their intersection to verify the value of that makes one-third of all orientation completers belong to the afternoon shift.
Calculate the conditional probability
Enter to calculate the favorable count divided by the count of volunteers not assigned to the evening shift. The exact answer is .
Step-by-step Explanation
Use the given conditional probability
The volunteers who completed the orientation consist of the morning, afternoon, and evening completion counts. Thus, the fraction of orientation completers assigned to the afternoon shift is
Find the missing table entry
Simplify and solve the equation:
So, .
Count the possible and favorable volunteers
Because the volunteer was not assigned to the evening shift, the possible volunteers are those assigned to the morning or afternoon shifts:
For the favorable count, include all afternoon volunteers and the morning volunteers who completed the orientation. This avoids counting the afternoon orientation completers twice:
Calculate the conditional probability
The required probability is
Therefore, the answer is .