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What does "at least one red" mean?
"At least one" means one or more. If you draw 2 marbles, "at least one red" includes:
In other words, everything EXCEPT drawing zero reds.
Why is complement the smart approach?
The hard way: Calculate P(exactly 1 red) + P(exactly 2 reds) separately.
The easy way: "At least one red" = "NOT zero reds". Just find P(no reds) and subtract from 1!
The Logic:
Every draw either has at least one red OR has no reds at all. These are complements!
P(at least one red) + P(no reds) = 1
The Key Formula:
A bag has 3 red and 5 blue marbles. Draw 2 marbles WITHOUT replacement. What's the probability of getting at least one red?
The Setup: 3 red + 5 blue = 8 total marbles
Important: What is "without replacement"?
When you draw a marble, you DON'T put it back before the next draw. So after the first draw, only 7 marbles remain. This changes the probabilities!
Find P(no red in 2 draws) — the complement
"No red" means both draws must be blue. Let's break it down:
Apply the complement rule
"At least one red" is everything except "no reds":
Result:
"At least one red" means:
• Case 1: First red, second blue
• Case 2: First blue, second red
• Case 3: Both red
Calculate each, then add them up!
"No red" means:
• Just one case: Both blue
One calculation, then subtract from 1!
Whenever you see "at least one", immediately think: complement! It's almost always easier to calculate the probability of getting NONE and subtract from 1, rather than adding up all the different ways to get one or more.