Relative frequency and experimental probability
Before a stickball game in the neighborhood, Wilkin flips a coin 4 times to decide who bats first and gets 3 heads. Does that mean the probability of heads is $0.75$? No: with only a few flips, chance can be surprising. If he flips it 1,000 times, the result gets much closer to one half.
Relative frequency of an event
For several outcomes, make a table with both frequencies. For example, a spinner spun 40 times:
| Color | Absolute frequency | Relative frequency |
|---|---|---|
| Red | 12 | $12/40 = 0.30$ |
| Blue | 10 | $10/40 = 0.25$ |
| Green | 8 | $8/40 = 0.20$ |
| Yellow | 10 | $10/40 = 0.25$ |
The relative frequencies add up to $1$ (or $100\%$).
A die was rolled $50$ times and the $6$ came up $8$ times. a) Calculate the relative frequency of the event "getting a $6$". b) Compare it with the theoretical…
A die was rolled $50$ times and the $6$ came up $8$ times.
a) Calculate the relative frequency of the event "getting a $6$".
b) Compare it with the theoretical probability $\frac{1}{6}$. Does the result seem reasonable to you?
The arrow of a spinner with $4$ colors was spun $40$ times, yielding these results: red $12$ times, blue $10$, green $8$, and yellow $10$. a) Make a table with the…
The arrow of a spinner with $4$ colors was spun $40$ times, yielding these results: red $12$ times, blue $10$, green $8$, and yellow $10$.
a) Make a table with the absolute and relative frequencies of each color.
b) Which color occurred with the highest relative frequency?
In a class of $30$ students, $12$ come to school by bicycle or scooter. a) What is the relative frequency of this event in the class? b) If the school has $600$…
In a class of $30$ students, $12$ come to school by bicycle or scooter.
a) What is the relative frequency of this event in the class?
b) If the school has $600$ students, how many would you expect to come by bike or scooter? Is this value exact or only an estimate? Why?
Relative frequency stabilizes with repetition
Roll the die many times and watch how the bars (what actually comes up) approach the orange line (what "should" come up according to Laplace):
Tossing a coin $200$ times resulted in $98$ heads. a) Calculate the relative frequency of the event "heads". b) Would you suspect that the coin is biased? Why?
Tossing a coin $200$ times resulted in $98$ heads.
a) Calculate the relative frequency of the event "heads".
b) Would you suspect that the coin is biased? Why?
A die is rolled an increasing number of times and the relative frequency of the event "getting a $4$" is recorded: after $10$ rolls it is $0.2$; after $50$, $0.18$;…
A die is rolled an increasing number of times and the relative frequency of the event "getting a $4$" is recorded: after $10$ rolls it is $0.2$; after $50$, $0.18$; after $100$, $0.16$; after $500$, $0.164$; after $1\,000$, $0.167$.
a) What value does the relative frequency seem to settle at?
b) How is this value related to Laplace's rule?
c) How many "fours" would you expect in $6\,000$ rolls?
The graph shows the relative frequency of "heads" when tossing a coin an increasing number of times. a) Read the relative frequency off the graph after $80$ tosses. b)…
The graph shows the relative frequency of "heads" when tossing a coin an increasing number of times.
a) Read the relative frequency off the graph after $80$ tosses.
b) What value does the relative frequency seem to settle at?
c) Do you think the coin is biased? Justify your answer.
The graph shows the relative frequency of "heads" in $500$ tosses of a coin. a) What value does the relative frequency settle at? b) Is the coin biased? Why? c) How many…
The graph shows the relative frequency of "heads" in $500$ tosses of a coin.
a) What value does the relative frequency settle at?
b) Is the coin biased? Why?
c) How many heads would you expect if it were tossed $1\,000$ more times?
Few repetitions can be misleading
With few trials, the relative frequency can stray far from the true probability, even though the probability has not changed.
A basketball player has an all-time free-throw percentage of $80\%$. In the last match, she scored only $3$ out of $5$. Her coach comments: "Her probability of scoring…
A basketball player has an all-time free-throw percentage of $80\%$. In the last match, she scored only $3$ out of $5$. Her coach comments: "Her probability of scoring has decreased today."
a) Do you agree with the coach? Justify your answer by distinguishing between probability and frequency over a small number of trials.
b) What would you expect if she took $100$ free throws in upcoming training sessions?
A trainer factory inspects $400$ pairs each day and records those with a defect: Monday $8$, Tuesday $5$, Wednesday $10$, Thursday $6$, and Friday $11$. a) Calculate the…
A trainer factory inspects $400$ pairs each day and records those with a defect: Monday $8$, Tuesday $5$, Wednesday $10$, Thursday $6$, and Friday $11$.
a) Calculate the relative frequency of defective pairs for each day.
b) Calculate the relative frequency of defective pairs for the whole week.
c) If a pair manufactured that week is chosen at random, what would you estimate the probability of it being defective to be?
A teacher asks students to spin a spinner with three sectors $360$ times: red ($180º$), blue ($90º$), and green ($90º$). Three students submit these results: | | red |…
A teacher asks students to spin a spinner with three sectors $360$ times: red ($180º$), blue ($90º$), and green ($90º$). Three students submit these results:
| red | blue | green | |
|---|---|---|---|
| Camila | $178$ | $96$ | $86$ |
| Luis | $120$ | $121$ | $119$ |
| Carla | $180$ | $90$ | $90$ |
a) Calculate the relative frequency of each color in the three tables.
b) What would be the expected frequencies based on theoretical probabilities?
c) Two of the students have cheated. Who are they and why?
After tossing a coin $4$ times and getting $3$ heads, Omar claims: "The probability of getting heads with this coin is $0.75$." a) Explain the error in his claim. b)…
After tossing a coin $4$ times and getting $3$ heads, Omar claims: "The probability of getting heads with this coin is $0.75$."
a) Explain the error in his claim.
b) What should Omar do to estimate the probability reliably?
c) If the coin is fair, what value will the relative frequency of heads approach as the number of tosses increases?
Compare with what is expected and decide
To find out whether a die, a coin, or a machine works the way it should, we compare the relative frequency with the theoretical probability.
It also helps when comparing two estimates: the one based on more repetitions is more reliable. If one team rolls two dice $30$ times and gets a sum of $7$ six times ($0.2$) and another team does it $300$ times and gets $50$ ($0.167$), the second estimate is better; moreover, the theoretical probability is $\dfrac{6}{36} = \dfrac{1}{6}$.
We suspect that a die is biased and we roll it $600$ times. The $6$ comes up $168$ times. a) Calculate the relative frequency of the event "getting a $6$". b) What…
We suspect that a die is biased and we roll it $600$ times. The $6$ comes up $168$ times.
a) Calculate the relative frequency of the event "getting a $6$".
b) What relative frequency would you expect if the die were fair?
c) Would you conclude that the die is biased? Why?
d) If we keep rolling it $300$ more times, how many "sixes" would you expect, based on your conclusion?
A supermarket has a bottle return machine that, on average, should not reject more than $2\%$ of valid containers. Over one week, the following data are recorded: | Day…
A supermarket has a bottle return machine that, on average, should not reject more than $2\%$ of valid containers. Over one week, the following data are recorded:
| Day | Containers | Rejected |
|---|---|---|
| Monday | $350$ | $9$ |
| Tuesday | $410$ | $8$ |
| Wednesday | $380$ | $12$ |
| Thursday | $420$ | $7$ |
| Friday | $390$ | $14$ |
a) Calculate the relative frequency of rejection for each day, rounding to three decimal places.
b) Calculate the relative frequency of rejection for the whole week.
c) The manager must decide whether to call technical support. What do you advise? Justify your decision using the data.
Two teams in a class experimentally estimate the probability of getting a total of $7$ when rolling two dice. Team A rolls $30$ times and gets this event $6$ times; Team…
Two teams in a class experimentally estimate the probability of getting a total of $7$ when rolling two dice. Team A rolls $30$ times and gets this event $6$ times; Team B rolls $300$ times and gets it $50$ times.
a) Calculate the relative frequencies for both teams.
b) Calculate the theoretical probability of "total $7$" by applying Laplace's rule.
c) Which estimate is more reliable? Why?
When Laplace's rule cannot be used
Sometimes we don't know whether the outcomes are equally likely (a thumbtack, a knucklebone, a jar lid when it lands). In those cases we experiment many times and use the relative frequency as an estimate of the probability.
A jar lid is flipped $100$ times and lands with the top facing up $64$ times. a) Estimate the probability that it lands face up. b) And that it lands face down? c) If it…
A jar lid is flipped $100$ times and lands with the top facing up $64$ times.
a) Estimate the probability that it lands face up.
b) And that it lands face down?
c) If it is flipped $500$ more times, how many times would you expect it to land face up?
d) Why can we not use Laplace's rule with the lid?
A knucklebone (a sheep bone used as a die) has been tossed $600$ times and landed with the flat side facing up $264$ times. a) What experimental probability would you…
A knucklebone (a sheep bone used as a die) has been tossed $600$ times and landed with the flat side facing up $264$ times.
a) What experimental probability would you assign to this outcome?
b) Explain why Laplace's rule cannot be applied in this experiment.
c) How many times would you expect it to land this way in the next $150$ tosses?
Simulating with random numbers
If you don't have a die or a coin on hand, you can simulate the experiment with a table of random numbers (or with the random function on a computer or phone). It is a list of digits from $0$ to $9$ in which each digit is equally likely.
You assign each outcome a number of digits proportional to its probability.
If an event has probability $0.3$, you assign it $3$ of the $10$ digits (for example, $0$, $1$, and $2$). The more digits you read, the better your estimate will be.