Relative frequency and experimental probability
Ximena flips a coin 4 times and gets 3 heads. Does that mean the probability of heads is $0.75$? No: with only a few flips, chance can surprise you. If she flips it 1,000 times, the result gets much closer to one half. Why does this happen, and how do we use it to estimate probabilities? Let's find out.
Relative frequency of an event
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 settles down as you repeat
Roll the die many times and watch how the bars (what actually comes up) get closer to the orange line (what "should" come up according to Laplace):
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?
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?
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.
Few repetitions can be misleading
With few attempts, the relative frequency can stray far from the real probability, even though the probability hasn't 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 | |
|---|---|---|---|
| Sofía | $178$ | $96$ | $86$ |
| Diego | $120$ | $121$ | $119$ |
| 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 the theory and decide
Sometimes you have to decide whether a result is "normal" or suspicious. The procedure is always the same: calculate the relative frequency, compare it with the theoretical probability, and think about how many repetitions there are.
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?
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?
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 can't be used
Sometimes we don't know whether the outcomes are equally likely (a thumbtack, a knucklebone die, a jar lid when it lands). In those cases you experiment many times and use the relative frequency as an estimate of the probability.
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?
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 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.
Back to the beginning: with 4 flips, $\frac{3}{4}$ is just a relative frequency from very few repetitions; to estimate the probability of heads you have to flip many times, and then the frequency gets close to $0.5$.