Statistics & Probability
Dependent Probability
Events are dependent when the outcome of the first affects the second. P(A then B) = P(A) × P(B|A), where P(B|A) means probability of B given A already happened.
Watch & Learn
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Worked Examples
Follow along with these step-by-step examples. Take your time — there's no rush!
Problem
A bag has 4 red and 6 blue marbles. Draw two WITHOUT replacement. P(red, then red)?
Step-by-Step Solution
P(1st red) = 4/10.
After removing one red: 3 red left, 9 total.
P(2nd red | 1st red) = 3/9 = 1/3.
P(red, red) = 4/10 × 3/9 = 12/90 = 2/15
Answer
P = 2/15
Problem
A deck has 52 cards. Draw two without replacement. P(ace, then ace)?
Step-by-Step Solution
P(1st ace) = 4/52 = 1/13.
After removing one ace: 3 aces, 51 cards.
P(2nd ace) = 3/51 = 1/17.
P = 1/13 × 1/17 = 1/221
Answer
P = 1/221
Problem
Bag: 3 green, 5 yellow. Draw two without replacement. P(green, then yellow)?
Step-by-Step Solution
P(green) = 3/8.
After removing green: 5 yellow, 7 total.
P(yellow | green) = 5/7.
P = 3/8 × 5/7 = 15/56
Answer
P = 15/56
Problem
Bag: 4 red, 4 blue. Draw two without replacement. P(red, then blue)?
Step-by-Step Solution
P(1st red) = 4/8 = 1/2.
After removing one red: 4 blue, 7 total.
P(blue | red) = 4/7.
P = 1/2 × 4/7 = 4/14 = 2/7
Answer
P = 2/7
Problem
Bag: 6 marbles (1 gold, 5 silver). Draw two without replacement. P(gold on 2nd draw | silver on 1st)?
Step-by-Step Solution
After drawing silver first: 1 gold, 4 silver remain (5 total).
P(gold | silver first) = 1/5.
Answer
P = 1/5
Your Turn — Practice Problems
Try all 5 problems on your own first. Write out your work — that's how it sticks!
💡 Tip: Don't peek at the answers until you've genuinely tried each one.
Bag: 5 red, 5 blue. Draw 2 without replacement. P(blue, blue)?
Bag: 2 green, 8 red. P(green, then red) without replacement?
Deck of 52. P(king, then king) without replacement?
Bag: 6 marbles (2 red, 4 blue). P(red, then blue) without replacement?
Why is drawing without replacement a dependent event?
Bag: 3 red, 7 blue. P(red, then red) without replacement?
Bag: 4 green, 6 yellow. P(green, then green) without replacement?
Deck of 52. P(ace, then king) without replacement?
Bag: 5 red, 5 blue. P(red, then blue) without replacement?
Bag: 10 marbles (1 gold). P(gold on 1st draw)?
Bag: 3 white, 7 black. P(white, then white) without replacement?
What makes two events dependent?
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