The Visits Hall Numbers
You are helping plan visits for a category C prison. The visits hall runs two sessions per day, five days a week. Each session can seat 24 prisoners, and each prisoner may have up to 3 adult visitors (children under 16 do not count toward the 24 prisoner places but do count toward the hall's legal maximum occupancy of 90 people at any one time).
Last week's data: 198 prisoners requested a visit. Of these, 41% brought 2 adults, 34% brought 1 adult, 15% brought 3 adults, and 10% brought no adult visitors (children only). The average number of children per visit was 1.2. The governor asks whether the hall can absorb a 15% increase in visit requests next month without adding sessions.
1. Calculate last week's total attendance (prisoners + adults + children) and the average per session. Show your working.
Read the worked answer
Prisoners: 198.
Adults:
- 41% × 198 = 81.18 → treat as 81 prisoners bringing 2 adults = 162 adults
- 34% × 198 = 67.32 → 67 bringing 1 adult = 67 adults
- 15% × 198 = 29.7 → 30 bringing 3 adults = 90 adults
- 10% × 198 = 19.8 → 20 bringing 0 adults = 0 adults
- Adults total ≈ 162 + 67 + 90 + 0 = 319 adults
Children: 198 visits × 1.2 = 237.6 ≈ 238 children
Total attendance ≈ 198 + 319 + 238 = 755 people.
Sessions: 2 per day × 5 days = 10 sessions. Average = 755 ÷ 10 = 75.5 people per session.
Reasoning: Round sensibly and state your rounding. The key check is that the average (75.5) is close to the occupancy cap of 90 - so the hall is already busy, and averaging alone hides peaks.
2. Can the hall absorb a 15% increase? Identify the binding constraint.
Read the worked answer
Prisoner places: a 15% rise gives 198 × 1.15 = 227.7 ≈ 228 prisoners over 10 sessions = 22.8 per session against a cap of 24. That is just within capacity, but with almost no slack - any clustering breaks it.
Occupancy: 755 × 1.15 ≈ 868 people over 10 sessions = 86.8 per session against a cap of 90. Again within, but only by about 3 people per session on average.
Binding constraint: the 90-person legal occupancy cap, because it is the tighter proportional margin once you account for clustering (a single session with several large family groups will breach 90 even if the average is fine).
Conclusion: on averages, yes - narrowly. In practice, no, not safely, without either spreading demand across sessions or capping visitors per prisoner. The honest answer to the governor is 'only if we manage peaks, not just totals.'
3. What single operational change would most improve headroom, and what is its cost?
Read the worked answer
Change: rebalance bookings so no session exceeds ~22 prisoners, moving requests to the quieter of the two daily sessions (and using the under-used days). This is demand management, not new capacity.
Alternative: reduce the adult allowance from 3 to 2 for the highest-demand weeks. That directly cuts the occupancy driver (3-adult visits = 90 adults from just 30 prisoners).
Cost: restricting visitors has a real rehabilitative and family-ties cost, and may be challengeable. Rebalancing is cheaper but depends on prisoners accepting different slots.
Reasoning: The exam rewards recognising that the constraint is peaks, and that the cheapest lever is scheduling, not building. State the trade-off rather than pretending there is a free fix.