Using Frequency tables and graphs to estimate probabilities
As we are beginning to see, probabilities occur in a wide variety of situations and we can visualise them through a wide variety of diagrams and tables.
Probabilities are also used in statistical analysis, usually as a way of predicting what might happen in the future based on past events that have been researched and recorded.
Obtaining probabilities from frequency tables
Our rule for probability is $\frac{\text{total favourable outcomes }}{\text{total possible outcomes }}$total favourable outcomes total possible outcomes
For example, the jump lengths of Long Jumpers in an athletics competition resulted in this data, we might then be interested in what the probability is that someone jumps more than $119$119cm.
and this will result in: total favourable = $8+4+1=13$8+4+1=13 and total possible is $31$31
$\frac{\text{total favourable outcomes }}{\text{total possible outcomes }}=\frac{13}{31}$total favourable outcomes total possible outcomes =1331 ≈ $42%$42%
We can use frequency tables quite easily to answer a range of questions with regards to probabilities.
Our rule for probability is $\frac{\text{total favourable outcomes }}{\text{total possible outcomes }}$total favourable outcomes total possible outcomes
Be careful of the language more than, less than, at least or at most. They are subtly different. For example for numbers from 1 to 10, the set {at least 4} contains elements {4,5,6,7,8,9,10} but the set {more than 4} contains the elements {5,6,7,8,9,10}.
Obtaining probabilities from graphs
This graph depicts data resulting from watching the number of people in a queue at a concert ticket stand and how long they waited in line to be served.
Let's look at how we calculate the probability of the following $2$2 questions.
a) What is the probability that someone was served in less than $40$40 minutes?
b) What is the probability that someone was served in a time of at least $40$40 minutes?
From the graph we can see that less than $40$40 minutes includes the number of people served in $30$30 minutes ($30$30 people) and the number of people served in $35$35 minutes ($60$60 people).
We will need the total number of people served, we can find this by adding up all the values of the columns
$30+60+110+50+30+10+10=300$30+60+110+50+30+10+10=300
The
| $\text{P(time to service < 40) }$P(time to service < 40) | $=$= | $\frac{\text{total favourable outcomes }}{\text{total possible outcomes }}$total favourable outcomes total possible outcomes |
| $=$= | $\frac{60}{300}$60300 | |
| $=$= | $\frac{1}{5}$15 | |
| $=$= | $20%$20% |
b) Now the question of "served in a time of at least $40$40 minutes" means find
$\text{P(time to service > 40) }$P(time to service > 40)
This is the complementary event for $\text{P(time to service < 40) }$P(time to service < 40) which we have already worked out. So the answer for this is
| $\text{P(time to service }$P(time to service $>=$>= $\text{40) }$40) | $=$= | $1-\text{P(time to service < 40) }$1−P(time to service < 40) |
| $=$= | $1-0.2$1−0.2 | |
| $=$= | $0.8$0.8 | |
| $=$= | $80%$80% |
We can use graphs quite easily to answer a range of questions with regards to probabilities, things to remember:
- Our rule for probability is $\frac{\text{total favourable outcomes }}{\text{total possible outcomes }}$total favourable outcomes total possible outcomes
- Take care when reading off information, check scales and check the axis labels.
Worked Examples
QUESTION 1
The following table shows the frequency of the lengths jumped at a long jump competition. The class interval is a range of distances measured in centimeters.
| Class Interval | Frequency |
|---|---|
| $0-39$0−39 | $6$6 |
| $40-79$40−79 | $9$9 |
| $80-119$80−119 | $7$7 |
| $120-159$120−159 | $8$8 |
| $160-199$160−199 | $1$1 |
| $200-239$200−239 | $2$2 |
| Sum | $33$33 |
According to the table, what is the probability that someone jumped more than 119 cm?
What is the probability that someone's jump measured between 80 and 159cm (inclusive)?
QUESTION 2
This frequency graph shows the number of people that were served at a furniture store, and the length of time it took to serve them.
What is the probability that someone was served in under $40$40 minutes?
What is the probability that someone had to wait at least $50$50 minutes to be served?