GED Mathematical Reasoning
Probability & Counting
Probability is a fraction: how many ways the thing you want can happen, over how many ways anything can happen. It is always between 0 (impossible) and 1 (certain). Counting problems are the same idea — you are just working out the bottom of that fraction.
The words first
- outcome
- one possible result — rolling a 4
- favorable outcomes
- the results you are asking about — the top of the fraction
- total outcomes
- every result that could happen — the bottom of the fraction
- independent events
- one does not affect the other, like two separate coin flips
The method
THE FRACTION
ways it CAN happen
probability = ——————————————————————
ways ANYTHING can happen
Rolling a 4 on one die: 1/6
Rolling an even number: 3/6 = 1/2 (2, 4, 6 all count)
TWO THINGS BOTH HAPPENING — multiply
two heads in a row: 1/2 × 1/2 = 1/4
ONE THING OR THE OTHER (they cannot both happen) — add
rolling a 2 or a 5: 1/6 + 1/6 = 2/6 = 1/3
WITHOUT REPLACEMENT — the bottom shrinks
2 red in a bag of 10, drawing two reds:
2/10 × 1/9 (one red gone, one marble gone)
Worked all the way through
The problem
A bag holds 4 red and 6 blue marbles. You draw one, keep it, then draw another. What is the probability both are red?
- Step 1 — First draw. Count the reds and the total.4 red out of 10 marbles → 4/10
- Step 2 — Now update the bag.You kept the red one. So there are 3 reds left and 9 marbles left.
- Step 3 — Second draw.3/9
- Step 4 — Both happening means multiply.4/10 × 3/9 = 12/90
- Step 5 — Simplify by dividing top and bottom by 6.12/90 = 2/15
On the board
4/10
first draw: 4 reds out of 10
3/9
second draw: one red and one marble are gone
4/10 × 3/9
both happening means multiply
12/90
2/15
divide top and bottom by 6
Answer
2/15
A second one, in a different form
A spinner has 8 equal sections: 3 red, 5 blue. What is the probability of red?
ways red can happen: 3
the top
ways anything can happen: 8
the bottom
3/8
Answer
3/8
The usual mistake
Writing 4 red / 6 blue as 4/6 instead of 4/10.
Check yourself
The trap is using 4/10 twice. Read for the word "keep" or "without replacing" — it tells you the bag changed between draws.
Practice it
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