Stats1 Chapter 4 (part 1). :: Probability

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Experimental

i.e. Dealing with collected data.

Theoretical

Deal with probabilities and modelling to make inferences about what we ‘expect’ to see or make predictions, often using this to reason about/contrast with experimentally collected data.

Chp1: Data Collection

Methods of sampling, types of data, and populations vs samples.

Chp2: Measures of Location/Spread

Statistics used to summarise data, including mean, standard deviation, quartiles, percentiles. Use of linear interpolation for estimating medians/quartiles.

Chp3: Representation of Data

Producing and interpreting visual representations of data, including box plots and histograms.

Chp5: Probability

Venn Diagrams, mutually exclusive + independent events, tree diagrams.

Chp6: Statistical Distributions

Common distributions used to easily find probabilities under certain modelling conditions, e.g. binomial distribution.

Chp7: Hypothesis Testing

Determining how likely observed data would have happened ‘by chance’, and making subsequent deductions.

Chp4: Correlation

Measuring how related two variables are, and using linear regression to predict values.

This chapter is a recap of the concepts you learnt at GCSE.

“I throw two fair die. Calculate the probability the sum of the two dice is more than 6.”

1 :: Basic Probability

2 :: Venn Diagrams

“Out of 50 students, 12 play both piano and drums, 30 play piano and 25 play drums. Find the probability a randomly chosen student plays neither instrument.”

3 :: Mutually Exclusive/Independent Events

4 :: Tree Diagrams

“The probability I hit a target is 0.3. If I hit it, the probability I hit again on the next shot is 0.4. If I miss, the probability I hit on the next shot is 0.1. If I shoot 3 times, what’s the probability I hit on the first and third shot?”

🖉 An experiment is a repeatable process that gives rise a number a number of outcomes.

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🖉 An event is a set of one or more of these outcomes.� (We often use capital letters to represent them)

🖉 A sample space is the set of all possible outcomes.

Two fair spinners each have four sectors numbered 1 to 4. The two spinners are spun together and the sum of the numbers indicated on each spinner is recorded.

Find the probability of the spinners indicating a sum of

(a) exactly 5 (b) more than 5

1 2 3 4

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4

2 3 4 5

3 4 5 6

4 5 6 7

5 6 7 8

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Spinner 1

Spinner 2

If the sample space is the amalgamation of two underlying experiments, a table is a helpful way to list the outcomes.

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The table shows the times taken, in minutes, for a group of students to complete a number puzzle.

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A student is chosen at random. Find the probability for a group of students to complete a number puzzle

(a) In under 9 minutes (b) in over 10.5 minutes.

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Pearson Statistics 1

Pages 56-57

We can either put frequencies or probabilities into the Venn Diagram.

0.6

0.85

0.25

0.15

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Dr Frost’s cat “Pippin”

Fro Tip: Start from the centre frequency and work your way outwards using subtraction.

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The following shows the results of a survey on the types of exercise taken by a group of 100 people.

65 run 48 swim

60 cycle 40 run and swim

30 swim and cycle 35 run and cycle 25 do all three

(a) Draw a Venn Diagram to represent these data. (4)

Find the probability that a randomly selected person from the survey

(b) takes none of these types of exercise, (2)

(c) swims but does not run, (2)

4. takes at least two of these types of exercise. (2)

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Jason is one of the above group. Given that Jason runs,

(e) find the probability that he swims but does not cycle. (3)

Jan 2012 Q6

Fro Tip: You’ll lose a mark if you don’t have a box!

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Pearson Statistics 1

Pages 59-60

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Example

1 2 3 4

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Fro Note: Independence does not affect how the circles interact in a Venn Diagram.

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Pearson Statistics 1

Pages 62-63

At GCSE we saw that tree diagrams were an effective way of showing the outcome of two events which happen in succession.

(Personal opinion however is that their use is easily avoidable)

There are 3 yellow and 2 green counters in a bag. I take two counters at random. Determine the probability that:

1. They are of the same colour.
2. They are of different colours.

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b

I like to list out the matching sequences of outcomes first, then find the probability for each.

The probability I hit a target on each shot is 0.3. I keep firing until I hit the target. Determine the probability I hit the target on the 5^th shot.

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Pearson Statistics 1

Pages 65-67

Extension Questions

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