Years from now, after a distinguished career in public service, Minister becomes Foreign Minister of .
Today, however, the you in front of the screen will play a different role to that revered minister: the minister's Chief of Staff. You will be weighing options from your staffers who are well-intentioned but not thoroughly familiar with statistics.
Minister is due at the Harvard Kennedy School for a summit, and everything runs through your desk. The numbers backing your staffers' strategic options sound right. But whether each one serves the situation is yours to judge.
BRIEFING 1 OF 3, TRAVELDECISION REQUIRED
The flights.
Minister is in a remote part of the world, far from any hub, and must reach the Kennedy School for the opening session. Ministry policy sets the bar:
For a critical event, the risk of the Minister arriving late must stay under 15%.
Direct charter
$50,000
One flight, no transfers. The flight has a 90% chance of landing on time.
Commercial
$8,000
Three flights with two transfers. Each leg of the flight has a 90% chance of landing on time.
Your staff crowds into your office. Three of them speak up.
Three options
MEMO, BUDGET OFFICERESPONSE REQUIRED
Justify the spend.
You chose the $50,000 charter over the $8,000 commercial route. Within the hour, the budget office replies:
Budget office"Every flight on both itineraries carried the same 10% risk of arriving late. You spent $42,000 more for identical odds. In a sentence or two, justify the charter before we release the funds."
Write at least a sentence before sending.
One strong version of the argument: "Commercial requires all three legs to arrive on time: 0.9 × 0.9 × 0.9 is about 73%, so the risk of arriving late is 27%, well over our 15% limit. The charter is a single flight with a 10% risk, which is under the limit." If your reasoning pointed in the same direction, your instinct was correct. The next screen provides the numbers.
CABLE, CAMBRIDGE MAON ARRIVAL DAY
The concept
The multiplication and complement rules
Multiplying the chances of events that must all happen is the multiplication rule. And the risk of arriving late is 1 minus the chance of arriving on time: the complement rule.
The bar below is every possible commercial trip. Each leg keeps 90% of the trips still on time. Tap × 0.9 once per leg and watch the on-time share shrink:
ALL COMMERCIAL TRIPS100%
red = risk of arriving late. it must not pass this line, the 15% limit.
each tap = one more leg that must arrive on time
27% is nearly double the 15% the policy allows.
We assume the three legs are independent: one delay does not cause the next. Real travel is more complicated, and we examine that assumption in class.
Minister 's arrival day could be disrupted by a transit strike or a storm. Your desk has the forecasts:
Transit strike
30%
Storm
40%
Both, on the same day
10%
Note the last row is given to you directly. You could not have multiplied 30% × 40% to get it — that would assume strikes and storms are independent, and here they are not.
Whenever the risk of disruption passes 65%, ministry rules require a contingency package, which in this case costs $12,000: a car and driver on standby, plus rooms held at the venue. Below that line, the Minister travels as planned.
The Minister asks you"Before we decide on the package, give me the odds. What is the chance my arrival day is disrupted at all?"
Three options
CABLE, CAMBRIDGE MACONTINGENCY DESK
The concept
The addition rule
30% + 40% = 70% would be right only if a strike and a storm could never share a day. They can, 10% of the time, and those days sit in both risks:
Add, then subtract the overlap once: 30% + 40% − 10% = 60%. If the two could never coincide, the overlap would be 0 and simple addition would be right.
Addition rule: P(A or B) = P(A) + P(B) − P(A and B). If A and B can't happen together, the overlap is 0 and you simply add.
BRIEFING 3 OF 3, THE ROOMQUESTION FROM THE MINISTER
Reading the room.
Minister has landed. Before the first session, your intelligence desk profiles every foreign minister in the room:
Ministers in the room
100
Friendly to
40
Signed the treaty
30
Of those 30 signers, friendly to
24
The Minister asks you"If a minister here is friendly to us, how likely are they to have signed the treaty?"
Give the Minister a number
CABLE, CAMBRIDGE MATHE ROOM
The concept
Conditional probability: which group are you looking at?
Here is everything you were told:
100 ministers in the room
40 friendly to
30 signed the treaty
24 of the signers are friendly (so 24 are both friendly and signed)
The number 24 appears in both answers. Divide it by a different group and you get a different result:
All 100 ministers, in the same layout as the probability table exercise. The 24 who are both friendly and signed sits in the top-left.
Signed
Did not
Total
Friendly
24
16
40
Not friendly
6
54
60
Total
30
70
100
Read across the Friendly row: 24 of 40 signed, 60%. Read down the Signed column: 24 of 30 are friendly, 80%. The same 24 in the corner, divided by its row total or its column total.
80% = 24/30: the group of signers. 60% = 24/40: the group of friendly ministers, which is what the Minister asked about. Same 24 people, different group.
Conditional probability: P(A given B) = P(A and B) / P(B). The group after "given" is the group you divide by, and P(A given B) is not P(B given A).
MINISTRY OF FOREIGN AFFAIRS, DEBRIEF
The week, totaled.
The flightsMultiplication & complement. All three legs must arrive on time: 0.9 × 0.9 × 0.9 ≈ 0.73. The Minister arrives late the rest of the time: 1 − 0.73 = 0.27.P(A and B and C) = P(A) × P(B) × P(C) · P(not A) = 1 − P(A)
The disruptionAddition. "A or B" is A plus B minus the overlap: 30% + 40% − 10% = 60%.P(A or B) = P(A) + P(B) − P(A and B)
The roomConditional probability. 24 of 30 signers are friendly (80%), but 24 of 40 friendly ministers signed (60%). The group you divide by decides the answer.P(A | B) = P(A and B) / P(B)
Note: a good decision and a good outcome are two different things. Take the flights: it is entirely possible to make the right call, book the charter, and have the Minister miss the opening session anyway.
Airport idea
The idea we hope you still remember if one of us from the teaching team runs into you at an airport in five years:
Articulate the numbers presented to you, and think through them systematically.
API-201THE SEMESTER AHEAD
What is to come.
Three ideas from this exercise that will get pushed further this semester.
IndependenceWe multiplied 0.9 × 0.9 × 0.9 as though each leg were independent: one delay never making the next more likely. Real travel is not so obliging. A storm closes a hub and all three legs suffer at once. We loosen that assumption later.
Bayes' ruleBriefing 3 only described the room. Conditional probability also updates: learn that a minister signed the treaty and the chance they are friendly moves from 40% to 80%. New information, revised belief. That machinery is Bayes' rule.
Decision analysisBriefing 1 asked only whether the risk cleared the policy. It never asked whether $42,000 was worth cutting the risk from 27% to 10%. Put probabilities and payoffs on the same tree and that question has an answer.