Chance basics
Probability as favourable outcomes over all outcomes.
Pen and paper is fine · no calculator needed why?
Opens at level 18. You're level 1. You can read and practice here now.
the idea
A probability is a share: the outcomes you want over all the equally likely outcomes. A bag of 3 red and 5 blue marbles gives red a chance of 3/8.
The outcomes must be equally likely. Two dice make 36 equally likely pairs, not 11 equally likely sums: six pairs make a sum of 7, and only one makes 2.
For "at least one", use the opposite. Work out the chance it never happens, then take that from 1.
techniques
Wanted over total
- Count every outcome: all the marbles, not only the other colours.
- Count the ones you want. For "not red", count everything that is not red.
- Write wanted over total and reduce.
worked example
A bag has 4 red, 6 blue and 2 green marbles. You pick one at random. What is the probability it is not blue? Answer as a fraction.
- 4 + 6 + 2 = 12 marbles.
- Not blue: 4 + 2 = 6.
- 6/12 = 1/2.
Answer: 1/2
Count the 36 pairs
- Two dice make 6 × 6 = 36 equally likely pairs.
- A sum of 7 has six pairs, and each step away from 7 loses one. Doubles are six pairs.
- For "at least", add the pairs for each sum, or subtract the rest from 36.
worked example
You roll two fair six-sided dice. What is the probability that the sum is at least 10? Answer as a fraction.
- Sums 10, 11 and 12: 3 + 2 + 1 = 6 pairs.
- 6/36 = 1/6.
Answer: 1/6
Use the opposite
- Find the chance of failing on one try: no 6 on a die is 5/6.
- Multiply that chance once for every try.
- Subtract the result from 1.
worked example
You flip a fair coin 3 times. What is the probability of getting at least one head? Answer as a fraction.
- No heads: 1/2 × 1/2 × 1/2 = 1/8.
- At least one head: 1 − 1/8 = 7/8.
Answer: 7/8
watch out for
- Dividing by the other colours instead of by everything. With 5 blue out of 12, the chance of blue is 5/12, not 5/7.
- Treating the 11 possible sums of two dice as equally likely.
- Adding chances for "at least one". Two rolls at 1/6 each do not make 2/6, because that counts a double six twice.
practice
One event
worked example
A bag has 4 green and 8 red buttons. You pick one at random. What is the probability it is green? Give the answer as a fraction in lowest terms.
Answer: 1/3
- There are 4 + 8 = 12 buttons, and 4 of them are green.
- So the probability is 4/12 = 1/3.
Two dice
worked example
You roll two fair six-sided dice. What is the probability that the sum is at least 11? Give the answer as a fraction in lowest terms.
Answer: 1/12
- Sums 11 to 12 come from 2 + 1 = 3 of the 36 equally likely pairs.
- So the probability is 3/36 = 1/12.
At least one
worked example
You roll a fair six-sided die twice. What is the probability of getting at least one 6? Give the answer as a fraction in lowest terms.
Answer: 11/36
- Use the opposite. The chance of no 6 in 2 rolls is (5/6)² = 25/36.
- So the chance of at least one 6 is 1 − 25/36 = 11/36.