L2 Probability
Question Bank
The time students spend commuting to school is normally distributed with an average time of $28 \text{ minutes}$ and a standard deviation of $5 \text{ minutes}$.
(a) Find the probability that a randomly selected student spends less than $20 \text{ minutes}$ commuting.
(b) Find the probability that a student spends between $30 \text{ minutes}$ and $40 \text{ minutes}$ commuting.
The diameter of tomatoes harvested from a large garden plot is normally distributed with an average diameter of $7.2 \text{ cm}$ and a standard deviation of $0.6 \text{ cm}$.
(a) Find the probability that a randomly selected tomato has a diameter greater than $8.0 \text{ cm}$.
(b) Find the probability that a tomato has a diameter between $6.5 \text{ cm}$ and $7.5 \text{ cm}$.
The waiting time for a customer service agent is normally distributed with an average wait time of $3.5 \text{ minutes}$ and a standard deviation of $0.9 \text{ minutes}$.
(a) Find the probability that a customer waits longer than $5.0 \text{ minutes}$.
(b) Find the probability that a customer's wait time is NOT between $3.0 \text{ minutes}$ and $4.0 \text{ minutes}$.
The total charge life of a specific brand of electric scooter batteries is normally distributed with an average life of 80 hours and a standard deviation of 7 hours.
(a) Find the probability that a randomly selected battery will last for less than 75 hours.
(b) Find the probability that a battery lasts between 78 hours and 90 hours.
(c) The manufacturer offers an extended warranty only to the top 15% of batteries. What is the minimum charge life required for a battery to qualify for this extended warranty?
The concentration of a specific contaminant in river water samples is normally distributed with an average level of 15.5 parts per billion (ppb) and a standard deviation of 2.1 ppb.
(a) A level below 12.0 ppb is considered safe. What is the probability that a randomly collected sample is considered safe?
(b) If the Environment Agency wants 99% of all samples to be below a certain limit ($L$), what is the value of this maximum allowable limit?
(c) If 800 water samples are tested, how many samples would be expected to have a contaminant concentration above 18.0 ppb?
The useful life of a heavy-duty tyre (measured in thousands of kilometers) is normally distributed with an average life of $58 \text{ thousand km}$ and a standard deviation of $8.0 \text{ thousand km}$.
(a) Find the probability that a randomly selected tyre will last between $50 \text{ thousand km}$ and $65 \text{ thousand km}$.
(b) Out of a sample of 250 tyres, how many would the company expect to last less than $48 \text{ thousand km}$?
(c) A fleet manager buys 4 new tyres for a vehicle. What is the probability that ALL FOUR of these tyres will last longer than $62 \text{ thousand km}$?
Two different coffee machines, Machine X and Machine Y, dispense coffee into large cups. The volume dispensed by both machines is normally distributed.
Machine X has an average volume of $350 \text{ ml}$ with a standard deviation of $8 \text{ ml}$.
Machine Y has an average volume of $355 \text{ ml}$ with a standard deviation of $4 \text{ ml}$.
(a) Find the probability that Machine X dispenses a volume less than $340 \text{ ml}$.
(b) If the capacity of the cup is $360 \text{ ml}$, which machine has the higher probability of overflowing the cup?
(c) Machine Y has a standard deviation that is half that of Machine X. Explain, using probability calculations, the practical difference this makes when comparing the chances of dispensing a volume less than $345 \text{ ml}$.
The weight of cereal in boxes filled by a machine is normally distributed with an average weight of 365 grams.
(a) Quality control data shows that 8% of the boxes contain less than 358 grams of cereal. Calculate the standard deviation of the cereal box weights.
(b) If the standard deviation remains constant, what is the probability that a randomly selected box contains a weight within 5 grams of the average?
(c) The machine must be adjusted if the heaviest 2% of boxes exceed a warning weight. What is this warning weight?
The final scores in a large statistics examination are normally distributed with an average score of 68 marks and a standard deviation of 11 marks.
(a) Find the probability that a student scores less than 60 marks.
The exam has the following awards. Students who score above 75 marks are awarded a distinction. Students who score above 80 are awarded high distinction, and students who are in the top 3% are awarded a special scholarship. Students can get only one award (the highest one they qualify for).
(b) What is the minimum score required to gain the special scholarship?
(c)What proportion of award winners gain high distinction?
The time taken for a pizza delivery service to reach the customer is known to be normally distributed.
(a) The company initially estimates the average delivery time as 35 minutes. If only 8% of deliveries take longer than 45 minutes, what is the standard deviation of the delivery times, based on this assumption?
(b) A new system is installed, and the standard deviation is controlled to be 5.0 minutes. If the company wants 95% of all deliveries to be completed within 40 minutes, what must the new average delivery time be?
(c) Using the average calculated in part (b) and a standard deviation of 5.0 minutes, what is the probability that a delivery takes between 30 and 38 minutes?
The heights of female adults are normally distributed with an average height of $168 \text{ cm}$ and a standard deviation of $6.5 \text{ cm}$.
(a) What is the probability that a randomly selected female adult is taller than $175 \text{ cm}$?
(b) When collecting data, the researcher noticed that $20\%$ of female adults were shorter than $163 \text{ cm}$. Use the average height of $168 \text{ cm}$ to calculate the standard deviation implied by this new data point.
(c) Further data gathering confirmed that $10\%$ of female adults are shorter than $160 \text{ cm}$, and $25\%$ are taller than $185 \text{ cm}$. Use this information to determine both the average height ($\mu$) and the standard deviation ($\sigma$) that best fit this entire data set.