L3 Probability Concepts
Question Bank
A school tennis player gets 70% of her first serves in. Assume that her serves are independent of each other.
(a) Calculate the probability that her next three first serves all go in.
(b) Calculate the probability that, of her next three first serves, the first two go in and the third does not.
In a family board game, a player must roll a 6 on an ordinary die before they can start moving their counter. They roll the die once on each turn.
Calculate the probability that a player has still not started after their first three turns.
At a school gala, a stall runs a dice game. A player rolls two ordinary dice and wins a prize if the total of the two dice is 9 or more.
(a) Calculate the probability that a player wins a prize.
(b) The stall expects about 180 games to be played during the gala. Approximately how many prizes should the stall expect to give away?
A Year 12 class has 25 students, and 10 of them take Chemistry. Two students are chosen at random from the class, one after the other, to help at a parents’ evening.
(a) Calculate the probability that the first student chosen takes Chemistry.
(b) The first student chosen takes Chemistry. Calculate the probability that the second student chosen also takes Chemistry.
A game at a holiday programme uses two spinners. Spinner A has three equal sections numbered 1, 2 and 3. Spinner B has four equal sections numbered 1, 2, 3 and 4. A player spins both spinners, and their score, $T$, is the total of the two numbers.
(a) Complete the probability distribution table for $T$.
| $t$ | $P(T = t)$ |
|---|---|
| 2 | $\dfrac{1}{12}$ |
| 3 | |
| 4 | |
| 5 | |
| 6 | |
| 7 | $\dfrac{1}{12}$ |
(b) A player wins a prize if their score is 6 or more. Calculate the probability that a player wins a prize.
A survey asked all 60 Year 13 students at a school in the Manawatū where they live and whether they hold a driver licence (restricted or full). The table shows the results.
| Licence | |||
|---|---|---|---|
| Lives | Yes | No | Total |
| Rural | 15 | 5 | 20 |
| In town | 16 | 24 | 40 |
| Total | 31 | 29 | 60 |
Two different students are chosen at random from those who live rurally. Calculate the probability that both of them hold a driver licence.
An ice-block company prints “WIN” on the stick of 1 in 5 of its ice blocks, spread at random through everything it makes. A winning stick can be swapped for a free ice block.
Its summer advertisement says: “Buy three ice blocks, and you’ll probably win at least once!”
Is the advertisement’s claim correct? Use calculations and statistical reasoning to support your answer.
A box of 12 Christmas crackers has a bonus prize inside 2 of the crackers. A family takes 3 crackers from the box at random.
Calculate the probability that at least one of the family’s 3 crackers has a bonus prize.
A hardware store in Gisborne finds that 20% of its customers buy paint. Three customers are chosen at random.
(a) Calculate the probability that exactly one of the three customers buys paint.
(b) Calculate the probability that two or more of the three customers buy paint.
In a game called “Six or Out”, a player rolls an ordinary die up to three times:
- A 6 wins the game and a 1 loses it. Either way, the game stops.
- A 2, 3, 4 or 5 means the player rolls again.
- If there is still no 6 or 1 after three rolls, the game is a draw.
(a) Calculate the probability that the game is a draw.
(b) Calculate the probability that the player wins the game.
A school’s top hockey team has won 60% of its games over the past three seasons.
Estimate the probability that the team wins only one of its next three games.
State and justify any assumption(s) that you have made in calculating this probability.
Two flatmates use a dice game to decide who cooks dinner each night:
- In each round, both flatmates roll an ordinary die.
- If the numbers are different, the flatmate with the lower number cooks.
- If the numbers are the same (a tie), they play another round.
(a) Show that the probability that a round is a tie is $\frac{1}{6}$.
(b) Over the last 60 nights, the flatmates have never needed more than three rounds to decide who cooks. Explain, using statistical reasoning, whether this is unusual.
In a Year 12 class of 20 students, 5 hold a first-aid certificate. Four students from the class are chosen at random to help at the school’s open day, and the organiser hopes that at least one of them has a first-aid certificate.
(a) Calculate the probability that at least one of the four students chosen holds a first-aid certificate.
(b) Another student calculated $1 - 0.75^4 = 0.6836$, using a probability of 0.25 for each of the four students chosen. Explain the assumption that this method makes, and discuss whether (or not) this assumption is likely to be valid here.