Spotting Conditional Statements in Talk

Lesson 1 of 2: Hearing the "Given" Group

In this lesson:

• Spot conditional statements in everyday language
• Explain why reversing the condition changes the meaning

Goals for This Two-Lesson Pair

By the end of this pair, you should be able to:

1. Spot conditional statements and their signal phrases
2. Explain why and differ
3. Describe independence in everyday terms
4. Translate real situations into informal probability statements
5. Critique flawed everyday probability claims

Among Triers, or Among Everyone?

Compare two claims:

• "75% of people who tried the app kept using it"
• "75% of everyone uses the app"

Not the same. The phrase "who tried it" restricts to a subgroup first.

Reword as "Chance of A Among B"

Every conditional claim fits one frame:

the chance of A among the group of B

• App claim: chance of kept using among those who tried
• Naming the subgroup reveals the conditioning

The "given" group is whatever follows "among," "who," or "if you are."

Phrases That Signal a Condition

These phrases flag a conditional statement:

Each restricts to a subgroup first. Hear one, then ask: which subgroup, and what chance?

The Subgroup Changes the Meaning

A conditional claim is narrower than an unconditional one:

• "75% of triers kept using it" — only about triers
• "75% of everyone uses it" — about the whole population

Same number, very different claims. The subgroup is everything.

Quick Check: Name the Given Group

Claim: "Among students who studied, 90% passed."

Name the given group and reword it as "chance of A among B."

Answer: given group = students who studied; chance of passing among those who studied.

Spotting the Group, Then Reversing It

You can now spot the "given" group. Now reverse it:

• "Chance of cancer if you smoke"
• "Chance of smoking if you have cancer"

These sound alike — but they ask different questions. Why?

Two Questions, Two Reference Groups

"Cancer given smoker" looks among smokers; "smoker given cancer" looks among cancer patients.

The Two Answers Can Differ Sharply

The two directions can land far apart:

• — often high
• — much lower

Different questions, different answers — even though they sound alike.

Base Rates Explain the Gap

Most people don't smoke; cancer is rare. Different-sized groups → the directions diverge.

The General Rule: Direction Matters

• Swapping the condition swaps the reference group
• "Most NBA players are tall" ✓ — "most tall people play in the NBA" ✗

Reversing a claim can flip it from true to absurd.

Your Turn: Reverse and Explain

Claim: "Most firefighters are brave."

• Write the reversed-condition version
• Explain why the two differ, and which is likely larger

Work both parts yourself. Think about the two reference groups.

Watch Out: Two Reasoning Errors

"A given B" = "B given A"? No — swapping reverses the question.

Base-rate neglect: when the condition is rare, the directions differ a lot.

Ask: which direction, and how common is the condition?

Key Takeaways and What's Next

✓ Signal phrases mark a chance within a subgroup
✓ Reword every claim as "chance of A among B"
✓ — direction matters

Reversing a claim can flip its meaning; watch base rates

Next: independence in words, and critiquing claims.