Events, probabilities, and dependence
Calculate delivery risks and identify when shared causes make an independence assumption misleading.
Calculate probabilities for simple business events and identify when treating events as independent would distort the result.
Overview
The owner of Pine Bakery orders essential packaging from two suppliers. Each has a 10% chance of missing Friday's deadline. She assumes the chance that both are late is just 1%. She needs to check whether their delivery risks are connected.
Both businesses and all probabilities in this lesson are fictional teaching assumptions.
Recall Lesson 035: Why did Larch Supply's downside scenario fail to establish a guaranteed minimum profit?
Define the event before the number
An event is a clearly defined occurrence, such as “Supplier A's complete shipment arrives after noon Friday.” A probability expresses how likely the event is, from zero to one, or 0% to 100%. Write 10% as 0.10 when multiplying.
The notation P(A) means “probability of event A.” A 10% chance of a minor delay and a 10% chance of losing a week's production have different business importance.
State the deadline, quantity, and conditions. Probabilities can come from relevant records or explicit judgments. A past frequency, such as ten late shipments out of 100, gives 10 ÷ 100 = 10%; using it for next Friday assumes the history is sufficiently comparable. It does not guarantee future counts.
The complement is everything outside the event. If a shipment is either late or on time, the probability it is on time is 1 − P(late). With a 10% late probability, that is 1 − 0.10 = 0.90, or 90%.
Independent and dependent events
Two events are independent when knowing that one occurred does not change the probability of the other. Different supplier names do not establish independence. They might share a carrier, warehouse, or weather exposure.
For independent events, multiply their probabilities to find the probability of both:
P(A and B) = P(A) × P(B).
If Pine's delivery failures were independent, both-late probability would be 0.10 × 0.10 = 0.01, or 1%. The probability that at least one shipment arrives on time would then be 1 − 0.01 = 99%.
When events are dependent, the probability of one changes with information about the other. Use a conditional probability: P(B given A), meaning the chance of B among situations where A happens. The general rule is P(A and B) = P(A) × P(B given A). OpenStax explains this multiplication rule and its simpler independent form. Source: OpenStax
Worked example
Pine learns that both suppliers use the same regional delivery hub. For this exercise, assume each supplier still has a 10% late probability, but B's chance of being late when A is late is 60%.
Both-late probability is therefore 0.10 × 0.60 = 0.06, or 6%. At least one on-time shipment has probability 1 − 0.06 = 94%, not 99%.
These model proportions per 100 comparable occasions are not observed records or guaranteed counts.
| Outcome | Occasions per 100 |
|---|---|
| Both late | 6 |
| Only A late | 4 |
| Only B late | 4 |
| Neither late | 86 |
The rows total 100. A is late on 6 + 4 = 10 occasions; B is also late on ten. Among A's ten late occasions, B is late on six, giving 6 ÷ 10 = 60%.
The owner understated the chance of losing both deliveries by five percentage points: 6% − 1%. Check the shared hub and alternative routes before treating a second supplier as effective protection.
For this example, either complete shipment alone is enough to operate. If the bakery needs both shipments, “at least one on time” answers the wrong question. Both on time has probability 86% in the table. Define operational success before choosing the calculation.
“Or” and “cannot happen together”
In probability, “A or B” usually includes both. To find the chance that at least one supplier is late, add their individual probabilities and subtract the overlap counted twice:
P(A or B) = P(A) + P(B) − P(A and B).
For Pine: 0.10 + 0.10 − 0.06 = 0.14, or 14%. This also equals 1 − 0.86.
Mutually exclusive events cannot both occur in the same defined situation. One shipment being late and that same shipment being on time are mutually exclusive. Two suppliers both being late are not. Independence and mutual exclusivity are different ideas, as OpenStax emphasizes. Source: OpenStax
With mutually exclusive events, there is no overlap to subtract. With two events that each have positive probability, mutual exclusivity also means they are not independent: knowing one happened rules out the other.
Exercise
Allow 8–10 minutes. You lead production at established manufacturer Cedar Instruments. A customer order ships Friday only if both a component delivery and a quality test are completed on time.
Let A mean the component delivery is late and B mean the test is late. Assume P(A) = 20%, P(B) = 15%, and P(B given A) = 50%. A late delivery can disrupt the test schedule. There are no other causes of a missed shipment in this exercise.
Management will accept the plan only if its modeled chance of shipping on time is at least 80%.
- Calculate the chance both activities are late. Compare it with the result from incorrectly assuming independence.
- Calculate the chance at least one is late, then the chance the order ships on time. Does the plan meet the requirement?
- Construct four outcome probabilities: both late, only A late, only B late, and neither late. Check that they total 100%.
- Explain why “not both late” is insufficient for this order. Name one assumption to verify and one action to investigate before promising delivery.
Try the exercise before reading on.
Answer
Both-late probability is 0.20 × 0.50 = 0.10, or 10%. Assuming independence would give 0.20 × 0.15 = 0.03, or 3%, understating simultaneous lateness by seven percentage points.
At least one late is 0.20 + 0.15 − 0.10 = 0.25, or 25%. Shipping on time requires neither to be late, so its probability is 1 − 0.25 = 0.75, or 75%. This misses the 80% requirement by five percentage points.
| Outcome | Probability |
|---|---|
| Both late | 10% |
| Only delivery late | 20% − 10% = 10% |
| Only test late | 15% − 10% = 5% |
| Neither late | 75% |
The four outcomes total 10% + 10% + 5% + 75% = 100%. Only the last outcome allows shipment. “Not both late” has probability 90%, but includes occasions when one necessary activity is late and the order still cannot ship.
Verify whether the conditional 50% estimate reflects the current delivery and testing schedule. Investigate earlier delivery or reserved testing capacity, then revise the probabilities using evidence. Check each action's effect and cost.
Dependence does not always move every risk measure in the same direction. Holding the individual late probabilities fixed, independence would give a 32% chance of at least one delay: 20% + 15% − 3%. The supplied dependence concentrates more delays together, producing 25% instead, while increasing simultaneous delays. Name the event you care about before saying risk rose or fell.
Recall answer: A downside scenario describes one possible result. It assigns neither a guaranteed bound nor a probability to every possible outcome. Today's calculations likewise depend on explicitly stated assumptions.
Applying this lesson
Write the event, deadline, probability source, and possible shared causes before combining probabilities. Keep the table for this week's decision tree. Next, we connect probabilities with the size of business gains and losses.
Sources
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