Relative frequency and experimental probability

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 $4$ colours 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 colour.

b) Which colour occurred with the highest relative frequency?

At a charity fair booth there is an opaque urn with winning and non-winning tickets. After opening the first $50$ tickets of the afternoon, the relative frequency of winning tickets has been exactly $0.14$.\n\na) How many winning tickets have been drawn in those first $50$ draws?\n\nb) The organizers add new tickets and aim for the total relative frequency of prizes to rise to exactly $0.20$ by the end of the next $150$ draws (reaching a cumulative total of $200$ tickets played). How many winning tickets would necessarily have to be drawn in that second batch of $150$ draws?\n\nc) Explain why, even if the organizers designed the urn with that intention, the random nature of the draw does not make it $100\,\%$ certain that exactly that calculated amount will be drawn in the second batch.

During this month's training sessions, Aoife has taken $80$ free throws at the basket and made $68$.\n\na) Without doing written division or using a calculator, estimate with reasoning whether her relative shooting accuracy is closer to $70\,\%$, $85\,\%$, or $95\,\%$. Explain what you focus on mentally to make your estimate.\n\nb) Calculate the exact value of the relative frequency of made shots expressed as an irreducible fraction, a decimal number, and a percentage. Check whether your estimate from the previous section was correct.\n\nc) In the next game, Aoife takes her first $4$ free throws and misses all $4$. A spectator claims: "Aoife has lost her touch; her probability of scoring has plummeted." Relying on the relationship between sample size and the stability of relative frequencies, explain why the spectator's conclusion has no mathematical basis.

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?

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 heads would you expect if it were tossed $1\,000$ more times?

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?

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 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?

Cian rolls a conventional $6$-sided cubic die a total of $12$ times to check if it is loaded. He gets the number $5$ on $4$ occasions. After the experiment, Cian states: "The relative frequency of rolling a $5$ is $4/12 = 1/3 \approx 0.33$. Since the theoretical probability is $1/6 \approx 0.17$, the die is clearly loaded. Furthermore, on the next roll it is almost impossible to get a $5$ again, because it has already come up too much."\n\na) Identify the two probabilistic reasoning errors Cian makes and explain why they are incorrect.\n\nb) Why does a series of only $12$ rolls not allow you to be sure whether the die is loaded or not?\n\nc) If the die were fair and rolled $1\,200$ times, around what value would you expect the relative frequency of getting a $5$ to stabilize, and approximately how many times would that number come up?

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:

a) Calculate the relative frequency of each colour 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?

A Met Éireann weather station in a coastal town has recorded the daily weather during the month of October over the past $20$ years, accumulating a historical series of $620$ October days. In that record, noticeable precipitation was recorded on $217$ of those days.\n\na) Model the probability of it raining on any given October day in that town based on its historical relative frequency. Express the result as an irreducible fraction, a decimal number, and a percentage.\n\nb) The first $4$ days of October this year have been completely dry and clear. A neighbor states: "Tomorrow, day 5, it is definitely going to rain to make up for it, because the probability of rain is high." Reason critically whether this statement is valid from the standpoint of chance.\n\nc) If the month of October has $31$ days, how many rainy days would theoretically be expected throughout the whole month according to your model? Does this mean that in any given year that exact number of days will necessarily have rain? Justify your answer using the concept of sample variability.

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?

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?

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 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?