Probability: practice problems
Probability comes down to counting: how many equally likely outcomes are there, and how many of them satisfy the condition?
Basic, Standard, Advanced: Grade 8 · Term 3
Math problem generator
Finding a probability
Treat every ball or ticket as different (red 1, red 2, ...) so that all outcomes are equally likely.
Two dice
Two dice give outcomes. A table makes sums and products easy to count. A sum of 7 happens in 6 ways, so its probability is .
Tree diagrams and common mistakes
Use tree diagrams for coins, cards and lotteries. Drawing two at once ignores order; drawing one after the other counts order.
- "At least one head" .
- The chance of winning a lottery does not depend on the drawing order.
- Always reduce the fraction.
Worked examples
A bag contains 7 red and 5 white balls. One ball is taken out. Find the probability that it is white.
Hint
Treat every ball as different; then each ball is equally likely.
Answer
Solution
There are 12 equally likely outcomes.
5 of them satisfy the condition.
So the probability is
A large die and a small die are thrown together. Find the probability that the product of the numbers is odd.
Hint
Two dice give outcomes. Make a table and count.
Answer
Solution
There are 36 equally likely outcomes.
9 of them satisfy the condition:
So the probability is
A bag contains 3 red and 3 white balls. A ball is taken out, its colour noted, and it is put back; then a ball is taken out again. Find the probability that both balls are the same colour.
Hint
The ball is put back, so the second draw is from the same balls. There are (number of balls) × (number of balls) outcomes.
Answer
Solution
There are 36 equally likely outcomes.
18 of them satisfy the condition.
So the probability is
Practice problems
One card is drawn from 15 cards numbered 1 to 15. Find the probability that the number is at least 12.
Hint
Count all equally likely outcomes and the ones that satisfy the condition.
Answer
Solution
There are 15 equally likely outcomes.
4 of them satisfy the condition:
So the probability is
There are 5 lottery tickets, 2 of them winners. A and B each draw one ticket in that order, without putting it back. Find the probability that both win.
Hint
Number the tickets so they are all different, and count the ways A and B can draw with a tree diagram.
Answer
Solution
With numbered tickets there are ways for A and B to draw.
2 of them satisfy the condition.
So
A large die and a small die are thrown; the large die shows and the small die shows . Let P be the point . Find the probability that P lies above the line .
Hint
Two dice give outcomes. Make a table and count.
Answer
Solution
There are 36 equally likely outcomes.
15 of them satisfy the condition:
So the probability is