In a valid model, the probabilities of all the possible outcomes add up to exactly 1. If they do not, something is missing, repeated or miscalculated. The total check is the quickest way to catch an error before you hand in an answer.
This lesson draws together mutually exclusive outcomes and tree diagrams, and it is the final check in probability reasoning.
What makes a model valid?
Three conditions must hold.
- Every probability is between 0 and 1.
- The outcomes cover everything that can happen and cannot overlap.
- The probabilities of all outcomes add up to 1.
The same idea gives you a missing probability: subtract the known ones from 1.
Worked example
A game has four results, A, B, C and D. The probabilities are P(A) = 2/5, P(B) = 1/4, P(D) = 1/10, and P(C) is unknown. Find P(C), then state P(B or C).
Step 1, common denominator. The denominators are 5, 4 and 10, so use 20. Then 2/5 = 8/20, 1/4 = 5/20 and 1/10 = 2/20.
Step 2, add the known ones. 8/20 + 5/20 + 2/20 = 15/20.
Step 3, subtract from 1. P(C) = 20/20 − 15/20 = 5/20 = 1/4.
Step 4, (B or C). B and C are mutually exclusive, so P(B or C) = 5/20 + 5/20 = 10/20 = 1/2.
Check. 8/20 + 5/20 + 5/20 + 2/20 = 20/20 = 1. The completed model is valid.
The mistake to watch for
Accepting a model because each value looks reasonable on its own.
A student models a biased die with P(1) = 0.1, P(2) = 0.2, P(3) = 0.3, P(4) = 0.2, P(5) = 0.1 and P(6) = 0.2.
Mistaken working: “Every probability is between 0 and 1, so the model is fine.”
Between 0 and 1 is necessary, but it does not make a model valid.
Correction. Add them: 0.1 + 0.2 + 0.3 + 0.2 + 0.1 + 0.2 = 1.1. The total is more than 1, so the model is not valid. At least one value is too large.
Totals also catch slips inside a tree. If the branches after one outcome are 0.7 and 0.2, they add to 0.9 and one branch is missing 0.1. A fast mental habit is to add each split as you draw it.
Check yourself
Try these, then open each answer.
1. A team’s results have P(win) = 0.45 and P(draw) = 0.25. There is no other result except a loss. Find P(loss).
Show answer
P(loss) = 1 − 0.45 − 0.25 = 0.30. Check: 0.45 + 0.25 + 0.30 = 1.00.
2. On a tree, the first branch is P(R) = 0.4. After R, the branches for the second stage are 0.7 and 0.2. What is wrong, and what must be true?
Show answer
The second-stage branches from one point must add to 1, but 0.7 + 0.2 = 0.9. One branch is missing 0.1 (either 0.8 and 0.2, or 0.7 and 0.3). The exact fix depends on the information given.
3. The probabilities of four outcomes are 3x, 2x, x and 4x. Find x and the probability of the outcome with 4x.
Show answer
3x + 2x + x + 4x = 10x = 1, so x = 0.1. The outcome with 4x has probability 0.4. The four probabilities are 0.3, 0.2, 0.1 and 0.4, which total 1.
Where this leads next
Now put all five skills together in the probability reasoning practice set, where each question is worked in full. The probability tree and counting board shows the total for any tree you build, and the non-calculator working trainer practises exact fraction sums like those above.
A total check takes seconds, yet it is skipped more often than any other step. If you would like a teacher to build it into your routine, online one-to-one Mathematics tuition is one way to do that.