How New Evidence Changes Prior Probabilities: A Simple Case Study

[Rajeev Bagra]

How New Evidence Changes Prior Probabilities: A Simple Case Study

Conditional probability problem
byu/DigitalSplendid inlearnmath

Here is a clean, publication-ready version of the explanation written in plain English, with no LaTeX or mathematical symbols, suitable for blogs, Medium, WordPress, or LinkedIn articles.

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From 50–50 to 10 out of 11: How One Clue Changes Everything

The situation

A crime is committed by one of two suspects, A or B.
At the start, there is no reason to prefer one over the other, so each has a 50 percent chance of being guilty.

Later, investigators discover an important fact:

• The guilty person has a rare blood type found in only 10 percent of the population.
• Suspect A has this blood type.
• Suspect B’s blood type is unknown.

The question is: how does this information change what we believe?

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Part (a): What is the probability that A is guilty?

Think in terms of how well the evidence fits each suspect.

If A is guilty

The evidence fits perfectly.
We already know A has the rare blood type, so the discovery tells us nothing new in this case.

If B is guilty

For the evidence to be true, B must also have the rare blood type.
But only 10 percent of people have it, so this situation is much less likely.

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Comparing the two possibilities

Imagine 100 similar cases:

• In 50 cases, A is guilty.
  In all 50 of those, the blood type evidence would appear.

• In 50 cases, B is guilty.
  Only about 5 of those would show the rare blood type.

So out of 55 cases that match the evidence,

• 50 involve A
• 5 involve B

That means A accounts for 50 out of 55, which simplifies to 10 out of 11.

Conclusion:
After seeing the blood type evidence, the probability that A is guilty is 10 out of 11.

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Part (b): What is the probability that B has the same blood type?

This is the part that usually causes confusion.

We are now asking:

Given everything we know, how likely is it that B also has this rare blood type?

To answer this, we must consider two different situations.

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Situation 1: A is guilty

This happens 10 out of 11 times.

If A is guilty, B’s blood type is unrelated to the crime.
So B has the rare blood type purely by chance, which happens 10 percent of the time.

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Situation 2: B is guilty

This happens 1 out of 11 times.

If B is guilty, then B must have the rare blood type.
So in this case, the probability is 100 percent.

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Combining both situations

We now add the contributions from both cases:

• Most of the time, A is guilty, and B has a small chance of matching the blood type.
• A small fraction of the time, B is guilty, and the match is guaranteed.

When these are combined, the final probability is 2 out of 11.

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Why the answer is not simply “10 percent”

At first glance, it feels like B’s blood type should still be random.

But the key insight is this:

The blood type evidence is linked to guilt.

Even though B is unlikely to be guilty, if B is guilty, then the blood type match is certain.
That small but guaranteed possibility raises the overall probability above 10 percent.

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Final results (plain language)

• Probability that A is guilty: 10 out of 11
• Probability that B has the rare blood type: 2 out of 11

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Takeaway

Evidence doesn’t just tell us what is likely —
it changes how possibilities are weighted.

A small chance combined with certainty can matter more than intuition suggests.

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