Statistics & Probability
Independent Probability
Two events are independent if one does not affect the other. P(A and B) = P(A) × P(B). Example: flipping a coin and rolling a die are independent.
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Worked Examples
Follow along with these step-by-step examples. Take your time — there's no rush!
Problem
You flip a coin and roll a 6-sided die. What is P(heads AND rolling a 4)?
Step-by-Step Solution
P(heads) = 1/2. P(rolling 4) = 1/6.
Events are independent.
P(heads AND 4) = 1/2 × 1/6 = 1/12
Answer
P = 1/12
Problem
A bag has 5 red and 5 blue marbles. You draw one, replace it, then draw again. P(red, then red)?
Step-by-Step Solution
With replacement → independent events.
P(red) = 5/10 = 1/2 each time.
P(red AND red) = 1/2 × 1/2 = 1/4
Answer
P = 1/4
Problem
P(A) = 0.3 and P(B) = 0.4. If independent, find P(A and B).
Step-by-Step Solution
P(A and B) = P(A) × P(B)
= 0.3 × 0.4 = 0.12
Answer
P(A and B) = 0.12
Problem
Roll a die and flip a coin. What is P(rolling a 6 AND flipping tails)?
Step-by-Step Solution
P(rolling 6) = 1/6. P(tails) = 1/2.
Independent events: multiply.
P = 1/6 × 1/2 = 1/12
Answer
P = 1/12
Problem
A spinner has 4 equal sections (A, B, C, D). You spin it twice. What is P(A then B)?
Step-by-Step Solution
P(A) = 1/4. P(B) = 1/4. Spins are independent.
P(A then B) = 1/4 × 1/4 = 1/16
Answer
P = 1/16
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.
P(A) = 1/3, P(B) = 1/4, independent. Find P(A and B).
Flip two coins. P(heads AND heads)?
Roll two dice. P(1 AND 1)?
P(A) = 0.5, P(B) = 0.6, independent. P(A and B)?
Spin a 4-section spinner twice. P(section 1 both times)?
P(A) = 1/2, P(B) = 1/3, independent. P(A and B)?
Flip a coin and roll a die. P(tails AND 3)?
P(A) = 0.4, P(B) = 0.25, independent. P(A and B)?
Roll a die twice. P(6 AND 6)?
Spin a 5-section spinner twice. P(section 2 then section 4)?
P(A) = 0.8, P(B) = 0.5, independent. P(A and B)?
Flip three coins. P(all heads)?
Finished all 12? Give yourself a pat on the back — then check your work!
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