Working Backwards Probability

Working backward from given probabilities to find a missing count

These questions flip the usual probability setup. Instead of asking you to find a probability, they give you probabilities and ask you to find a missing count. The key insight: since all probabilities must add to 1, you can find the missing probability and then convert it to a count.

The Core Method

Given: total count $N$, and probabilities for all categories except one.

Step 1: Find the missing probability: $P(\text{missing}) = 1 - P(\text{given}_1) - P(\text{given}_2)$

Step 2: Convert to a count: $\text{count} = P(\text{missing}) \times N$

That's the entire technique. The probabilities of mutually exclusive, exhaustive categories must sum to exactly 1.

Example 1 — MCQ Format

A library's fiction section contains a total of $600$ books, categorized as mystery, science fiction, or fantasy. If a book is selected at random, the probability of selecting a mystery book is $0.35$, and the probability of selecting a science fiction book is $0.40$. How many books are categorized as fantasy?

A) 25
B) 150
C) 210
D) 30

Step 1 — Find P(fantasy): $1 - 0.35 - 0.40 = 0.25$
Step 2 — Convert to count: $600 \times 0.25 = 150$
Answer: B

Example 2 — SPR (Fill-In) Format

A car dealership has a total inventory of 450 vehicles: sedans, trucks, or SUVs. If a vehicle is selected at random, the probability of selecting a sedan is $0.36$ and the probability of selecting a truck is $0.28$. How many SUVs are in the inventory?

Step 1 — Find P(SUV): $1 - 0.36 - 0.28 = 0.36$
Step 2 — Convert to count: $450 \times 0.36 = 162$
Answer: 162

Common Traps and Gotchas

• Confusing the probability with the count. Choice A in Example 1 is "25" — that's the percentage (25%), not the number of books. Always multiply the probability by the total.

• Giving a count for the wrong category. Choice C in Example 1 is 210, which is the number of mystery books ($600 \times 0.35$). Read the question carefully to make sure you're finding the right category.

• Subtracting only one probability. If you do $600 \times (1 - 0.35) = 390$, that gives the combined count of science fiction and fantasy — not fantasy alone. You must subtract all the given probabilities from 1.

• Arithmetic shortcuts. Since $P(\text{missing}) = 1 - P_1 - P_2$, you can do the subtraction first and multiply once. This is faster and less error-prone than computing each category separately and subtracting.

What to Do on Test Day

• Spot the pattern: You're given a total count and two (or more) probabilities. The question asks "how many" of the remaining category. That's your cue for working backwards.
• Subtract probabilities from 1 first, then multiply by the total. One multiplication is cleaner than two.
• Double-check by verifying the parts sum to the whole. In Example 1: $210 + 240 + 150 = 600$ ✓
• Key formula: $\text{missing count} = N \times (1 - P_1 - P_2)$
• All probabilities in a complete set sum to 1. This is the foundational fact that makes the entire technique work.