Probability Tree Diagrams
1) With Replacement
A bag contains 4 red sweets and 3 green sweets.
I take a sweet out of the bag, replace it and take a second.
Complete the probability tree diagram to show the different outcomes
Find: a) The probability of getting two red sweets 16/49
b) The probability of getting one red sweet 12/49 +12/49 = 24/49
c) The probability of getting at least one red sweet
16/49 + 12/49 +12/49 = 40/49
2) Without Replacement
A bag contains 4 red sweets and 3 green sweets.
I take two sweets out of the bag at the same time
Complete the probability tree diagram to show the different outcomes
Find: a) The probability of getting two red sweets 12/42
b) The probability of getting one red sweet 12/42 +12/42 = 24/42
c) The probability of getting at least one red sweet
12/42 + 12/42 +12/42 = 36/42
Bar Charts, Frequency Diagrams and Histograms
1) Bar Charts – only use these for discrete data
2) Frequency Charts – only use these for continuous data.
Time, s / Frequency0≤ t <5 / 8
5≤ t <15 / 20
15≤ t <35 / 30
35≤ t <40 / 12
A frequency chart has a continuous scale along the x axis. The bars show the
varying interval widths.
3) Histograms – this is the best graph for continuous data, especially when the
intervals are different widths.
A histogram has a continuous scale along the x axis. The bars show the
varying interval widths. The area of each bar shows the frequency.
The height of each bar is the Frequency Density = Frequency÷ Interval width
Time, s / Frequency0≤ t <5 / 8
5≤ t <15 / 20
15≤ t <35 / 30
35≤ t <40 / 12
Eg
Time, s / New Height0≤ t <5 / 8 ÷ 5 = 1.6
5≤ t <15 / 20 ÷ 10 = 2
15≤ t <35 / 30 ÷ 20 = 1.5
35≤ t <40 / 12 ÷ 5 = 2.4
Questionnaires and Surveys
A good question should be specific in what it is asking and it should provide response boxes
The response boxes should not overlap and should include options for all the possible responses.
Eg. To investiage pupils’ methods for purchasing music the following question was asked:
How much music do you buy?
0 – 3 pieces 3 – 5 pieces 5- 8 pieces
Criticise this question.
1) Options overlap!
2) No time given! The question should say how many items of
music/songs have you bought in the past month?
3) What if someone buys more than 8 pieces? Needs a “more than 8”
option box.
When doing a survey it is important to take a representative sample. If asked in the exam about a sampling technique, look for ways in which the sample might be biased. It is also important that the sample size is fairly big. A sample size of less than 30 is too small.
Eg A council wants to find opinions about a town. It does so by sending questionnaires to a single housing estate. Is this a good survey.
No. A single housing estate does NOT REPRESENT the whole town.
Should send to Several mixed housing estates
Eg A school wants to find out opinions about school uniform so it sends a questionnaire to 30 pupils in Year 10. Is this a good survey.
No. A single year group may be biased and does NOT REPRESENT
the whole school., should ask several pupils in each year group.
Eg. A magazine wants to find out what people think about its current content. They phone 200 readers on their land line in Huntingdon to see what they think. Is this a good survey.
No. Huntingdon is only a single town so opinions may be BIASED
it may NOT REPRESENT all of the readership. By phoning on the
land line you are only getting views from those people that are
available to answer the phone at home, this may result in BIAS.
Eg A supermarket wants to extend its store so it asks people in town during the day about current provision. Is this a a good survey technique?
No. People in town during the day may not be representative of all the
people that use the shop. What about people that work during the
day?
NOTE: Any survey technique that takes a sample can be criticised in some way – avoiding BIAS is almost impossible! So let the criticisms flow!
Scatter graphs
Describe the correlation:
As ______increases so does ______- A POSITIVE CORRELATION
As ______increases ______decreases – A NEGATIVE CORRELATION
CORRELATIONS THAT ARE TIGHTLY BUNCHED are said to be STRONG
Eg.
Look at the scatter graph opposite which shows
the length of time people took to travel from their
house into town against the distance of their house
from town.
a) Describe the correlation
Weak Positive
b) Draw a line of best fit
Make certain that the line follows the data and has roughly the
same number of crosses either side!
c) What time would you expect someone living 8 miles from town to take to get into town?
Around 29 minutes (reading up from 8 miles)
d) Why can’t you use the graph to predict the time for someone living 16 miles from town?
Because you can only reliably use the line of best fit to read of values
for those inside the range of the data collected. 16 miles is outside this
range.
Comparing two sets of Data
Finding the Mean, Median and Mode for data in a table
Eg. The data below shows how many goals were scored in several games
during a hockey tournament. Find the mean number of goals scored in
a game.
Number of Goals / Number of Games0 / 1
1 / 2
2 / 3
3 / 5
4 / 6
5 / 4
Find the mean length of a worm
Number of Goals / Number of Games0 / 1 / 0 1 = 0
1 / 2 / 1 2 = 2
2 / 3 / 2 3 = 6
3 / 5 / 3 5 = 15
4 / 6 / 4 6 = 24
5 / 4 / 5 4 = 20
Mean = 67 21 = 3.2
What is the median value?
middle one is the 11th counting down the number of games the 11th is a game with 3 goals scored
(11 items of data: 0 1 1 2 2 2 3 3 3 3 3)
What is the mode?
4 because it has the highest frequency (second column)
Eg The data below shows how long people had to wait to have their phone call
answered on a help line.
Wait Time, t / Number of people0 < t < 1 / 5
1 < t < 3 / 6
3 < t < 5 / 4
5 < t < 11 / 3
11 < t < 20 / 1
Find an estimate for the mean ‘wait time’. We can only estimate the actual mean time because the data has bee n grouped so we don’t have the actual raw values.
Wait Time (mins), t / Number of people / mid point0 < t ≤ 1 / 5 / 0.5
1 < t ≤ 3 / 6 / 2
3 < t ≤ 5 / 4 / 4
5 < t ≤ 11 / 3 / 8
11 < t ≤ 20 / 1 / 15.5
19 70
Mean = 70 ÷ 19 = 3.7 (1 d.p.)
Which interval will the median lie in?
Middle one is the 10th as 19 items of data. This is in the 1 < t < 3 interval
What is the modal class?
1 < t < 3 as it is the interval with most in (6)
Stem and Leaf Diagrams
1) Remember to order the data
2) No commas between the numbers
3) Remember to do a key!
4) Remember to space the numbers equally along each stem in columns
– No Crossing Out!!
Probability
1) Remember to use fractions (or decimals if the question is using them)
NOT RATIOS
2) Remember that the probability something doesn’t happen is found by
subtracting the probability that it does happen from 1.
Eg. If I roll a die the probability that I get a six is 1/6
The probability that I don’t get a six is 5/6
(because 1/6 + 5/6 = 1 whole)
Eg. The probability that it rains tomorrow is 0.35
The probability that it doesn’t rain tomorrow is 0.65
(because 0.35 + 0.65 = 1 whole)
3) The probability that two or more things happen one after the other can be
found by multiplying the probabilities together.
Eg. The probability that Mr Banham forgets to combs his hair is 3/4 on a
any given day.
The probability that Mr Banham forgets to comb his hair on two
days in a row = 3/43/4 = 9/16
The probability that Mr Banham forgets to comb his hair on three
days in a row = 3/43/4 3/4 = 27/64
The probability that Mr Banham forgets to comb his on just the middle
day in a three day spell = 1/43/41/4 = 3/64
The probability that Mr Banham forgets to comb his hair just once in a
three day spell= 3/41/41/4 = 3/64 (forgets on day 1 only)
= 1/43/41/4 = 3/64 (forgets on day 2 only)
= 1/41/43/4 = 3/64 (forgets on day 3 only)
So P(forgetsjust once in a 3 day spell) = 3/64 + 3/64 + 3/64 = 9/64
Relative Probability
This is when we use an experiment to estimate the probability of an outcome.
Relative probability of an outcome = Number of times that outcome happened
Total number of observations made
Score on Dice / Frequency1 / 3
2 / 4
3 / 2
4 / 5
5 / 4
6 / 3
Eg
Note that if you make more observations and repeat the experiment more times the value of the relative probability will be more accurate.
Eg. I want to calculate the probability of me being held up at a set of traffic lights on the way to work. In each 5 day working week I counted up how many times I was stopped at the traffic lights.
WeekNo. of times held up
1 1
2 2
3 1
4 0
5 3
6 1
7 2
8 1
What is the relative probability of me being held up at the traffic lights after 1 week? Answer: 1/5 = 0.2
What is the relative probability of me being held up at the traffic lights after 2 weeks? Answer: 3/10 = 0.3
What is the relative probability of me being held up at the traffic lights after 3 weeks? Answer: 4/15 = 0.267
What is the relative probability of me being held up at the traffic lights after 8 weeks? Answer: 11/40 = 0.275
The most reliable value for the probability is 0.275
Finding an Expected Result using Probability
If you know the probability of a result you can use it to predict how many times you will get the result in the future.
Eg. A die was rolled lots of times and the relative probability of getting a six
was found to be 0.2. If I roll the same die 200 times, how many sixes would I expect?
200 × 0.2 = 40 (simply multiply the number of rolls by the probability)
Frequency Polygon
Draw these for data that has been grouped. The polygon shows the ‘shape’ of the distribution. Draw the frequency polygon by plotting the frequency at the midpoint of each group.
Eg.
BOX PLOTS
CUMULATIVE FREQUENCY CURVES