I think of the difference as:

joint probability: refers to whole sample space

conditional probability: refers to restricted sample space (only among those who satisfy the condition)

However, I encountered this question (which is about constructing a tree diagram) and this really sounds like conditional probability instead of joint probability and would like to ask why this cannot be a conditional probability.

78% of students can construct tree diagrams. Of those who can construct tree diagrams, 97% passed, while only 57% of those students who could not construct tree diagrams passed.

Of those means those who constructed tree diagrams so 'able to construct a tree diagram' should be a condition. However, when calculating this fact is regarded as 'joint probability' that is P(tree diagram = yes AND passed = yes) instead of (passed = yes | tree diagram = yes). Maybe I don't understand English well but that phrase really sounds like conditional probability.

How do I effectively differentiate these two concepts from the sentence?


Let $A$ be the event of "the student can construct a tree diagram", and $B$ be the event of "the student passed".

You are told $\mathsf P(A)=0.78, \mathsf P(B\mid A)=0.97, \mathsf P(B\mid A^\complement)=0.57$

One clue confirming that these values are indeed for conditional probabilities is that a joint probability cannot exceed the value of either marginal probability.   Ie: $\mathsf P(A\cap B)\leq \mathsf P(A)$, but $0.97\not\leq 0.78$ so clearly $0.97\neq\mathsf P(A\cap B)$.

However, $\mathsf P(A\cap B)~{=\mathsf P(A)\,\mathsf P(B\mid A) \\= 0.78\cdot0.97\\=0.7566\qquad\qquad\leq 0.78}$


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