At the beginning of the year, a teacher seats his students alphabetically to learn their names quickly. Can we use a probability distribution to model where a particular child will sit in the class? Explain your answer
A manager randomly selects four people from his Sales team and two people from his Development team to attend a leadership conference. Can the number of people the manager picks to attend the leadership conference be modelled by a probability distribution? Explain your answer.
A manager randomly selects three of his staff to attend a leadership conference. He randomly selects people from his Sales team and his Development team.
Explain why the number of Sales people chosen can be modelled by a discrete probability distribution.
Given that there are four people on the Sales team, list all of the possible outcomes of this distribution.
The traffic signal at a pedestrian crossing is red for 3 minutes. Frank has just reached the pedestrian crossing and the light is red.
Explain why the length of time Frank will be waiting for the light to turn green can be modelled by a continuous probability distribution.
Sketch a graph that best models the shape of this continuous probability distribution. Let t represents the time Frank could wait at the light and P \left( t \right) represents the probability that Frank will wait for time t.
On average, the number of customers driving through Khalid's Koffee to buy a coffee is 50 an hour.
Explain why the length of time between customers can be modelled by a continuous probability distribution.
Sketch a graph that best models the shape of this continuous probability distribution.
The mass of each egg in a 12-pack carton of Extra Large Eggs ranges in weight from 66\text{ g} to 72\text{ g}.
Explain why the mass of the eggs in the carton can be modelled by a continuous probability distribution.
Sketch a graph that best models the shape of this continuous probability distribution.
A teacher asks her class to construct a triangle using a protractor and a ruler. She has given them a diagram with a 10 \text{ cm} base length and a base angle of 90 \degree. She asks them to draw a second base angle of 50 \degree. Some of the students aren’t very accurate with their drawing and the third angle at the apex does not necessarily measure 40 \degree exactly.
Explain why the size of the third angle at the apex can be modelled by a continuous probability distribution.
Sketch a graph that best models the shape of this continuous probability distribution.
When people are diagnosed with a Stage 3 lung cancer, their life expectancy tends to range between six months and two years. Around 15\% of patients live longer than 5 years.
Explain why the life expectancy of a patient diagnosed with Stage 3 lung cancer can be modelled by a continuous probability distribution.
Sketch a graph that best models the shape of this continuous probability distribution.
The average number of years that a couple remain married in Croatia, where the divorce rate is 60\%, is 14 years.
Explain why the number of years a couple is married in Croatia can be modelled by a continuous probability distribution.
Sketch a graph that best models the shape of this continuous probability distribution.
The weights of gold samples mined from a gold mine in Johannesburg are collected.
Explain why the weight of the gold samples can be modelled by a continuous probability distribution.
Sketch a graph that best models the shape of this continuous probability distribution.
The weight of gold samples mined from the the entirety of Johannesburg are collected and averaged. Sketch a graph that best models the shape of this continuous probability distribution.
The data given shows the heights of a group of 16 year-olds to the nearest centimetre:
| \text{Heights (cm)} |
|---|
| 148, 161, 154, 160, 150, 153, 155, 158, 156, 168 |
| 147,157,153,165,148,162,164,163,154,154 |
Complete the following relative frequency table:
| Height | Frequency | Relative Frequency |
|---|---|---|
| 145\leq h\lt 150 | ||
| 150\leq h\lt 155 | ||
| 155\leq h\lt 160 | ||
| 160\leq h\lt 165 | ||
| 165\leq h\lt 170 | ||
| 170\leq h\lt 175 |
Construct a relative frequency histogram.
Calculate the probability of a student being between 155 and 159 \text{ cm} tall, inclusive.
Calculate the probability of a student being less than 155 \text{ cm}.
The data below shows the time spent waiting for a green light at a set of traffic lights, in seconds:
| Time (seconds) |
|---|
| 11, 120, 5, 74, 32, 90,31,66,91,32.978 |
| 81,37,94,17,84,101,56,41,14,52 |
Complete the following relative frequency table:
| Time | Frequency | Relative Frequency |
|---|---|---|
| 0\leq t\lt 10 | 1 | |
| 10\leq t\lt 20 | ||
| 20\leq t\lt 30 | ||
| 30\leq t\lt 40 | ||
| 40\leq t\lt 50 | ||
| 50\leq t\lt 60 | 2 | |
| 60\leq t\lt 70 | ||
| 70\leq t\lt 80 | 1 | |
| 80\leq t\lt 90 | 2 | |
| 90\leq t\lt 100 | ||
| 100\leq t\lt 110 | ||
| 110\leq t\lt 120 |
Construct a relative frequency histogram.
Calculate the probability of a person waiting being between 20 and 29 seconds, inclusive.
Calculate the probability of a person waiting less than 30 seconds.
The IQ test results for 50 people aged 30 is represented by the relative frequency histogram below. The range 85 \leq x \lt 90 is represented by the number 90 on the x-axis:
Calculate the probability of a 30 year old having an IQ between 95 and 109 inclusive.
Calculate the probability of a 30 year old having an IQ between 100 and 130.
Consider the cumulative frequency ogive that has been drawn for the continuous data set in the table below:
| x | f |
|---|---|
| 1100 \text{ to } 1300 | 2 |
| 1300 \text{ to } 1500 | 3 |
| 1500 \text{ to } 1700 | 1 |
| 1700 \text{ to } 1900 | 6 |
| 1900 \text{ to } 2100 | 13 |
| 2100 \text{ to } 2300 | 10 |
| 2300 \text{ to } 2500 | 4 |
| 2500 \text{ to } 2700 | 5 |
| 2700 \text{ to } 2900 | 3 |
| 2900 \text{ to } 3100 | 3 |
Calculate the following probabilities to two decimal places:
P\left(X \gt 1100\ \vert\ X \lt 1900\right)
P\left(X \gt 1700\ \vert\ X \lt 2100\right)
The average time (t), in seconds, between 50 customers filling up their cars at a petrol station on a Monday morning between 8 am and 10 am is given in the relative frequency histogram below:
The results in the range 0 \leq t \lt 20 are represented by the number 20.
Calculate the probability of between 0 and 19 seconds passing, inclusive, between customers.
Calculate the probability of a customer waiting between 80 and 139 seconds.
A random digit generator generates 50 random digits between 0 and 999 inclusive. The results are shown in the relative frequency histogram below:
The range 0 \leq x \lt 100 is represented by the number 100 on the x-axis. Calculate:
P(X\lt 200)
P(X\geq 400)
P(X\lt 800| X\geq 400)
P(X\geq 500|X\lt700)
The number of minutes spent watching YouTube videos per day was monitored for a group of 50 fifteen year olds. The data is given below in a relative frequency histogram:
The range 105 \leq x \lt 115 is represented by the number 115 on the x-axis. Calculate:
P(X\lt 165)
P(X\geq 125)
P(X\lt 155| X\geq 125)
P(X\geq 145|X\geq 105)
The time, in seconds, between customers arriving at an express checkout is gathered.
Complete the table by finding the relative frequencies for each class interval:
| Class intervals | Relative frequency | Cumulative frequency |
|---|---|---|
| 0-5 | 0.29 | |
| 5-10 | 0.52 | |
| 10-15 | 0.68 | |
| 15-20 | 0.83 | |
| 20-25 | 0.9 | |
| 25-30 | 0.95 | |
| 30-35 | 1 |
If 200 customers were monitored, how many waited less than 20 seconds?
If 200 customers were monitored, how many waited between 15 and 20 seconds?
Describe the skewness of the data.
Calculate, up to two decimal places, the probability that, of the next 4 customers, the first 2 wait more than 20 seconds and the last two do not.
Calculate, up to two decimal places, the probability that, of the next 4 customers, exactly 2 wait more than 20 minutes.
The number of hours that a particular brand of light bulb lasts for is given in the frequency table below:
| Hours | Frequency | Relative Frequency | Cumulative Relative Frequency |
|---|---|---|---|
| 675 \text{ to } 725 | 0 | ||
| 725 \text{ to } 775 | 1 | ||
| 775 \text{ to } 825 | 1 | ||
| 825 \text{ to } 875 | 1 | ||
| 875 \text{ to } 925 | 6 | ||
| 925 \text{ to } 975 | 14 | ||
| 975 \text{ to } 1025 | 12 | ||
| 1025 \text{ to } 1075 | 7 | ||
| 1075 \text{ to } 1125 | 4 | ||
| 1125 \text{ to } 1175 | 4 | ||
| 1175 \text{ to } 1225 | 0 | ||
| 1225 \text{ to } 1275 | 0 | ||
| 1275 \text{ to } 1325 | 0 |
Complete the relative frequency and cumulative relative frequency columns in the table.
Hence, sketch a cumulative frequency ogive.
The cumulative frequency for a set of continuous data is given below:
| Hours | Cumulative Frequency | Cumulative Relative Frequency |
|---|---|---|
| 725\leq t\lt 775 | 1 | |
| 775\leq t\lt 825 | 1 | |
| 825\leq t\lt 875 | 5 | |
| 875\leq t\lt 925 | 14 | |
| 925\leq t\lt 975 | 29 | |
| 975\leq t\lt 1025 | 35 | |
| 1025\leq t\lt 1075 | 40 | |
| 1075\leq t\lt 1125 | 44 | |
| 1125\leq t\lt 1175 | 47 | |
| 1175\leq t\lt 1225 | 48 | |
| 1225\leq t\lt 1275 | 49 | |
| 1275\leq t\lt 1325 | 50 |
Complete the cumulative relative frequency column in the table.
Hence, calculate the following probabilities:
P(X\geq 825)
P(X\lt 1025\ \vert\ X\geq 875)