Use conditional probabilities and independence with positive denominators.
Loading your saved responses…
Try
A bag contains three red and two blue balls. Draw two without replacement. Before calculating, predict how seeing a red first changes the chance of red second. It removes one red from a smaller bag, so the new probability is2/4 rather than3/5. The observations are linked by the sampling mechanism.
Predict
Compare this with replacing the ball and thoroughly remixing before the second draw. Under the usual independent-draw model, the second red probability remains3/5. Predict which procedure makes two reds more likely, and explain the mechanism rather than relying on a memorized rule.
Experiment
Without replacement, P(two reds)=(3/5)(2/4)=3/10. With independent replacement, it is(3/5)(3/5)=9/25. Draw two probability trees and label each second branch conditional on the first result.
Now use a table example: among100 people,20 are club members,40 are runners and10 are both. P(runner given member)=10/20=1/2, while P(member given runner)=10/40=1/4. Shade the conditioned group first. Both calculations use the same overlap but different reference groups.
Understand
For an event B with positive probability, conditional probability is P(A given B)=P(A and B)/P(B). It asks what fraction of B also belongs to A. The restriction P(B)>0 matters: the elementary ratio is undefined when its denominator is zero. More advanced conditioning on continuous quantities uses additional machinery and is not obtained by dividing zero by zero.
Rearranging gives the multiplication rule P(A and B)=P(B)P(A given B). This rule does not require independence. Independence is the special case where P(A given B)=P(A), producing the product P(A)P(B). With positive relevant probabilities, the corresponding reverse conditional equality follows too.
Use the model, not the narrative order alone, to judge independence. Two draws may be dependent because of removal; two devices may fail together because of a common environment. Conversely, events can be independent in a stated probability model even if their descriptions share vocabulary. Verify the probability relation or justify the mechanism.
A finite empirical table can show conditional proportions that match exactly or nearly. That is descriptive evidence about the sample, not a proof that population events are independent. Inference about association requires an appropriate sampling design and procedure. Do not confuse a model equality supplied in a mathematics problem with an estimated equality from noisy data.
Pairwise independence of several events is weaker than joint independence. Knowing every pair satisfies the product rule does not automatically permit multiplying probabilities for a three-way intersection. This lesson uses two-event checks and explicitly independent repeated trials; broader probability courses develop the distinction with counterexamples.
Conditioning is also not automatically a causal intervention. People who choose a treatment can differ from people randomly assigned to it. P(outcome given treatment) summarizes an observed conditional relationship unless the design supports a causal interpretation.
Apply
Explain the two bag probabilities in words, then calculate a third-draw branch after two reds without replacement. Only one red remains among three balls, so that conditional probability is1/3. For the club table, reverse a conditional question deliberately and check that your answer changes with the denominator.
Continue
Next combine conditional branches to update beliefs. Bayes’ rule is bookkeeping over the same joint outcomes, with base rates kept visible.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
P(A and B)=3/20 and P(B)=1/4. Find P(A given B).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A bag has3 red and2 blue balls. Draw two without replacement. Find P(both red).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
With the same bag, replace and remix the first ball. Find P(both red).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
If P(B)=0, the elementary ratio definition P(A given B) is:
A table gives P(A)=0.4, P(B)=0.5, and P(A and B)=0.2. Check independence and explain why this is a model statement, not proof of population independence from a small sample.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Give an example where P(A given B) differs from P(B given A), and calculate both.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
