In this lesson:
By the end of this pair, you should be able to:
We're about to watch that total shrink as we condition.
A bag has 10 marbles: 4 red, 3 blue, 3 green. You draw one.
Does that change the chance it's red? You just gained information.
If it's not green, only 7 marbles remain (4 red, 3 blue).
The sample space shrank from 10 to 7 — conditioning recounts in the smaller world.
Conditioning on makes the new sample space.
Zoom into ; ask what fraction of it is also . Outcomes outside don't count.
"What fraction of 's probability also has ?" — the zoom written in symbols.
30 students; restrict to band (8 students), of whom 4 play sports.
Counting within the band row and the formula agree.
Given , , .
Find before advancing.
The formula doesn't care where the numbers come from:
One move every time: find and , then divide.
200 people; restrict to coffee drinkers (120), 40 of whom drink tea.
Among coffee drinkers, one in three also drinks tea.
and generally differ — the denominators differ.
Same joint cell, different totals — order changes the answer.
Both exceed the plain probabilities — rain and traffic are dependent.
Different denominators, different answers — one formula.
Regions: A only , overlap , B only , neither .
Work all four yourself. The given event sets each denominator.
? No — divide by ; it's within .
? No — different denominators, different answers.
Whatever follows the bar is the given event — and the denominator.
✓ — zoom into ✓ Read it from tables, statements, or Venns alike ✓ in general
The bar has a direction — given event is the denominator
Next: independence and medical testing.
The bar has a direction — given event is the denominator