Probability and statistics
Choose the correct sample space and distinguish frequency, density and averages.
What you will learn
- Calculate combined and conditional probabilities.
- Calculate weighted means and histogram frequencies.
Update the sample space
Without replacement, both the total number of objects and the number of a selected colour change after a draw. Multiply probabilities along one possible path, then add probabilities of mutually exclusive successful paths.
A conditional probability restricts the sample space. If a student is chosen from musicians, the denominator is the number of musicians rather than the whole class. Count the successful cases within that restricted group.
Frequency is a weight
For a frequency table, multiply each value by its frequency, add the products, and divide by the total frequency. Averaging the listed values without using frequencies gives the wrong weight to rare and common values.
In a histogram, area represents frequency. Therefore a bar's frequency is its class width multiplied by its frequency density. Unequal class widths require proportionally unequal bar widths in the drawing.
Worked example
A bag holds 4 red and 2 blue counters. Two are drawn without replacement. Find the probability of the same colour.
Show the worked solution
- Both red: (4/6)(3/5)=12/30.
- Both blue: (2/6)(1/5)=2/30.
- Add the disjoint cases: 14/30.
Answer 7/15
Common mistakes
- Without replacement is not the same as independent repeated trials.
- A histogram bar's height is density, not necessarily frequency.
What can you explain now?
Values 1, 2 and 4 have frequencies 2, 3 and 5. Find the mean.
Compare with the explanation
2.8
The weighted sum is 2+6+20=28 and the total frequency is 10.
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