Combined events
“And”, “or”, and drawing without putting back.
Pen and paper is fine · no calculator needed why?
Opens at level 24.
the lesson
Read the lesson
The idea
Combined events ask about two or more things at once. "And" means all of them happen. When the events don't affect each other (they are independent), like a coin and a die, multiply their chances: heads and a 6 is 1/2 × 1/6 = 1/12.
Draw without replacement and each pick changes what is left, so the next chance uses one fewer item.
"Or" means at least one happens. Count both groups, then take away the overlap, the outcomes in both, so nothing is counted twice.
Techniques
Multiply for "and"
- Find each event's chance on its own.
- Multiply the chances and reduce.
worked example
You flip a fair coin and spin a spinner with 4 equal sections numbered 1 to 4. What is the probability of tails and an even number? Answer as a fraction.
- Tails: 1/2. An even number: 2/4.
- 1/2 × 2/4 = 2/8 = 1/4.
Answer: 1/4
One fewer each draw
- First draw: the items you want over all the items.
- Next draw: one fewer of what you want, and one fewer in all.
- Multiply the fractions and reduce.
worked example
A bag has 3 red and 5 blue marbles. You draw 2 at random, one after another, without putting any back. What is the probability that both are red? Answer as a fraction.
- First red: 3/8.
- Second red: 2/7.
- 3/8 × 2/7 = 6/56 = 3/28.
Answer: 3/28
Add, then take off the overlap
- Count the outcomes in each group.
- Subtract the outcomes in both, which you counted twice.
- Divide by the total number of outcomes.
worked example
You pick a whole number from 1 to 20 at random. What is the probability it is even or a multiple of 5 (or both)? Answer as a fraction.
- Even: 10. Multiples of 5: 4. Both (10 and 20): 2.
- 10 + 4 − 2 = 12 numbers.
- 12/20 = 3/5.
Answer: 3/5
Tips by skill
- TipIndependent “and”: Independent events that must all happen: multiply their chances, then reduce.
- TipWithout replacement: After each draw, take one off both the count you want and the total. Multiply the fractions.
- Tip“Or” with overlap: Add the two counts, subtract the ones that are both, then divide by the total.
Watch out for
- Adding the chances for "and". All of them happening can never be more likely than any one of them.
- Keeping the same chance on every draw. Without replacement, each draw leaves one fewer.
- Counting the overlap twice. The king of hearts is a heart and a king, but it is one card.
skills · practice stats
-
Independent “and” not tried yet
worked example
You flip a fair coin and roll a fair six-sided die. What is the probability of heads and a multiple of 3 on the die? Give the answer as a fraction in lowest terms.
Answer: 1/6
- The results don’t affect each other, so multiply: 1/2 × 2/6 = 2/12 = 1/6.
-
Without replacement not tried yet
worked example
A bag has 8 orange and 10 white marbles. You draw 2 marbles at random, one after another, without putting any back. What is the probability that both are orange? Give the answer as a fraction in lowest terms.
Answer: 28/153
- 8/18 for the first, then 7/17 for the second: each pick leaves one fewer.
- 8/18 × 7/17 = 56/306 = 28/153.
-
“Or” with overlap not tried yet
worked example
You pick a whole number from 1 to 20 at random, each equally likely. What is the probability it is a multiple of 4 or a multiple of 10 (or both)? Give the answer as a fraction in lowest terms.
Answer: 3/10
- Multiples of 4: 5. Multiples of 10: 2. Both (multiples of 20): 1.
- 5 + 2 − 1 = 6 of the 20 numbers, so the probability is 6/20 = 3/10.
rest ladder
- 1 day
- 3 days
- 7 days
- 14 days
- 30 days
- 60 days
- mastered · every 90 days