Exercises: Explain Conditional Probability and Independence in Everyday Language
Work through each section in order. This standard is about EXPLAINING in
plain words, so several problems ask you to write complete sentences.
Remember: "A given B" and "B given A" are different questions with
different reference groups; "independent" means knowing one tells you
nothing about the other (it is about information, not whether the events
can happen together).
Warm-Up: Conditional and Independence Language
These problems review the plain-language meaning of conditioning and independence.
A news report says, "Among people who exercise daily, 80% sleep well." Which conditional probability does this claim describe?
P(sleeps well | exercises daily)
P(exercises daily | sleeps well)
P(sleeps well and exercises daily)
P(exercises daily)
In plain language, two events are independent when:
Knowing that one happened tells you nothing about the chance of the other.
They cannot both happen at the same time.
They always happen together.
One of them causes the other.
A coach says, "Most professional basketball players are tall." Rewrite the claim with the condition REVERSED by filling each blank. "Most ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ people are ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ players." (First blank: tall or short. Second blank: a profession.)
Fluency Practice
Identify the conditioning, judge independence, and read the table.
An ad claims, "9 out of 10 dentists who tried our toothpaste recommend it." Which paraphrase correctly restates the conditioning?
The chance of recommending it, restricted to the group of dentists who tried it.
The chance of trying it, restricted to the group of dentists who recommend it.
The chance that a randomly chosen person both tries and recommends it.
The chance that any dentist recommends it, whether or not they tried it.
A headline reads, "75% of people who tried the app kept using it." Which of the following is the UNCONDITIONAL "cousin" of this claim — a different, stronger statement about everyone?
75% of all people use the app.
75% of people who kept using the app had tried it.
Among people who tried the app, 75% kept using it.
25% of people who tried the app stopped using it.
A fair coin is flipped twice. Are the events "first flip is heads" and "second flip is heads" independent?
Yes — knowing the first flip tells you nothing about the second.
No — after a head, the next flip is more likely to be tails to balance out.
No — they cannot both be heads.
Yes — because they can never happen together.
A roulette wheel has landed on red five times in a row. A gambler says, "Black is due now, so black is more likely on the next spin." Is the gambler right?
No — spins are independent, so the next spin is unchanged by the past; black is not more likely.
Yes — after five reds, black must come up to balance the colors.
Yes — long runs always reverse on the next spin.
No — red is now more likely because it is on a streak.
A single card is drawn. Let $A = $ "the card is a heart" and $B = $ "the card is a spade." A student claims " and are independent because they cannot both happen." What is the best response?
They are NOT independent. "Cannot both happen" is mutual exclusivity; knowing the card is a heart tells you it is definitely not a spade, so one outcome gives information about the other.
The student is right; events that cannot co-occur are independent.
They are independent because they are different suits.
They are independent because each has the same probability.
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