How to Find Probabilities using Two-Way Frequency Tables?
Finding Probabilities using Two-Way Frequency Tables Example 1:
| Brown | Blue | Green | Total | |
| Male | 50 | 17 | 9 | 76 |
| Female | 43 | 14 | 6 | 63 |
| Total | 93 | 31 | 15 | 139 |
a) \(\frac{number \ of students \ with \ green \ eyes}{number \ of \ students}=\frac{15}{139}\)
Find Probabilities using Two-Way Frequency Tables: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- List outcomesName the possible results or count them carefully.
- Choose the ruleUse addition, multiplication, or conditional probability based on the wording.
- Check the rangeA probability must be between 0 and 1.
Worked examples
Simple probability
- Even outcomes are 2, 4, and 6.
- There are 3 favorable outcomes out of 6.
- Reduce the fraction.
Independent events
- P(heads) = 1/2 for each flip.
- The flips are independent.
- Multiply 1/2 by 1/2.
Try one before moving on
Find Probabilities using Two-Way Frequency Tables: pop-up practice
b) \(\frac{number \ of \ females}{number \ of \ students}=\frac{63}{139}\)
Finding Probabilities using Two-Way Frequency Tables Example 2:
Students were asked what their hair color was; the results are shown below. If a boy is selected at random, find the probability that the boy has brown hair.
| Black hair | brown hair | Blond hair | Total | |
| Boys | 32 | 13 | 3 | 48 |
| Girls | 26 | 16 | 9 | 51 |
| Total | 58 | 29 | 12 | 99 |
Solution: \(\frac{number \ of \ boys \ with \ brown \ hair}{number \ of \ students}=\frac{13}{99}\)
Exercises for Finding Probabilities using Two-Way Frequency Tables
Solve all the problems.
1) The two-way table shows the distribution of members of the audience at a play.
| Circle | Balcony | Stalls | Total | |
| Children | 32 | 64 | ||
| Adults | 20 | 13 | ||
| Total | 50 | 35 | 30 | 115 |
a) Complete the two-way tables.
b) What is the probability that a randomly chosen audience member is a child and is seated in the circle?
c) What is the probability that a randomly chosen audience member is a child?
2) The following table represents the data collected from 120 conference attendees of different nationalities:
| Arabic Speaker |
English Speaker | French Speaker | Total | |
| Man | 43 | |||
| Woman | 15 | 12 | 44 | |
| Total | 35 | 60 | 25 | 120 |
a) Complete the two-way tables.
b) Find the probability that a randomly selected participant is an English-speaking woman.
1)
| Circle | Balcony | Stalls | Total | |
| Children | 32 | \(\color{blue}{15}\) | \(\color{blue}{17}\) | 64 |
| Adults | \(\color{blue}{18}\) | 20 | 13 | \(\color{blue}{51}\) |
| Total | 50 | 35 | 30 | 115 |
b) Probability: \(\frac{32}{115}\)
c) Probability: \(\frac{64}{115}\)
2)
| Arabic Speaker |
English Speaker | French Speaker | Total | |
| Man | \(\color{blue}{20}\) | 43 | \(\color{blue}{13}\) | \(\color{blue}{76}\) |
| Woman | 15 | \(\color{blue}{17}\) | 12 | 44 |
| Total | 35 | 60 | 25 | 120 |
b) Probability: \(\frac{17}{120}\)
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