Relative frequency and experimental probability

We flipped a coin $50$ times and heads came up $28$ times.

1. a) What is the absolute frequency of "heads"? And of "tails"?
2. b) Calculate the relative frequency of "heads" as a fraction and as a decimal.
3. c) And the relative frequency of "tails"?

A basketball player has scored $45$ of her last $60$ free throws taken.

1. a) What is her relative frequency of scoring?
2. b) If she takes a free throw tomorrow, what experimental probability would you assign to her scoring it? Express it as a fraction, as a decimal and as a percentage.

At a container recycling plant in Cork, an optical sensor inspects plastic bottles to separate the defective ones. During a morning test, the sensor detected exactly $18$ defective bottles, which accounted for a relative frequency of defects of $0.03$.

1. a) State the corresponding mathematical relationship and calculate how many bottles the sensor inspected in total during that morning test.
2. b) How many non-defective bottles passed through the sensor in that test? What was their relative frequency expressed as a percentage?
3. c) If in the afternoon the machine checks a new batch of $800$ bottles of the same type and under identical conditions, how many defective bottles is it expected to find? Justify your answer using the concept of experimental probability.

Cian is playing with a fair $1$-euro coin. He flips it $10$ consecutive times and gets $8$ tails and only $2$ heads. When he finishes, Cian states very confidently: “Since the theoretical probability of getting heads is $0.5$, in the next $10$ flips at least $8$ heads must necessarily come up to compensate for the streak and so that the relative frequency stabilizes at the theoretical probability.”

1. a) Identify the error in Cian's statement and explain why his reasoning is false by recalling the principle that “chance has no memory”.
2. b) Calculate the relative frequency of the event “getting heads” in Cian's first $10$ flips as a fraction and as a decimal. Does this result contradict the fact that the coin is fair?
3. c) If Cian were to continue flipping the same coin until completing $1,000$ flips, explain how the stabilization of the relative frequency actually takes place: does the coin need to “give back” the missing heads or does the impact of those initial $8$ tails simply dilute among the large total number of flips?

A group of students rolls a die $60$ times and records the results: $1$ comes up $8$ times; $2$, $11$ times; $3$, $9$ times; $4$, $10$ times; $5$, $13$ times; and $6$, $9$ times.

1. a) Construct the table of absolute and relative frequencies (the latter to three decimal places).
2. b) What "theoretical" relative frequency would you expect for each face if the die is fair? Express it as a fraction and as an approximate decimal.
3. c) Which outcome was furthest from expected? Do you think the die is rigged? Justify your answer taking into account the number of rolls.

You can do it on paper and send a photo with your answer.

A board game club organizes a raffle with a spinner of $5$ equal sectors numbered from $1$ to $5$. Before the draw, a spectator tests it by spinning it $500$ times and gets: $1$ comes up $97$ times; $2$, $103$ times; $3$, $98$ times; $4$, $101$ times; and $5$, $101$ times.

1. a) Calculate the relative frequencies of each number.
2. b) What theoretical probability does each number have if the spinner is fair?
3. c) Does the spinner seem balanced? Reason your answer.

In the playground, students play dropping a drawing pin: it can land point up or on its side. After $200$ drops, the drawing pin landed point up $112$ times.

1. a) Calculate the relative frequency of "point up" and of "on its side".
2. b) What experimental probability would you assign to each outcome?
3. c) Why can we not use Laplace's rule in this experiment to say that each outcome is equal to $\frac{1}{2}$?

In a Mathematics class, students want to find out the probability that a solid wooden pyramid with a square base and four identical triangular faces lands resting on its square base when tossed into the air. To investigate this, two proposals arise:

• Aoife's method: Apply Laplace's rule directly, reasoning that, since the solid has $5$ faces in total and one of them is the base, the theoretical probability is $\dfrac{1}{5} = 0.2$ ($20\,\%$).

• Seán's method: Toss the pyramid $400$ times onto a cushioned surface, record how many times it lands resting on the base, and calculate the relative frequency of that event.

1. a) Compare both methods and justify which one is suitable for this experiment. What essential condition does Laplace's rule require that is not met in this geometric object?
2. b) When applying the correct method and tossing the pyramid $400$ times, the base lands resting on $144$ occasions. Assign the experimental probability to the events “resting on the base” and “resting on a triangular face”, expressing them as irreducible fractions, decimals, and percentages.
3. c) If a third student suggested tossing the pyramid only $10$ times to finish sooner, explain why their estimate would not be scientifically reliable compared to the one with $400$ tosses.

The graph shows how the relative frequency of the event "getting heads" changes as a coin is tossed more and more times.

1. a) Which value does the relative frequency approach as the number of tosses becomes very large?
2. b) Why do you think the fluctuations are large at the beginning and small at the end?
3. c) If we flipped the coin $100,000$ times, what relative frequency would you expect to get, approximately? And how many heads?
4. d) Does this graph contradict the statement "in $10$ tosses, $5$ heads must come up"? Reason your answer.