Exercises: 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.

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.

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 $\frac{1}{2}$?

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.

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.

Two basketball players boast about their accuracy:

• Lara has scored $12$ of her last $20$ free throws.

• Miriam has scored $45$ of her last $100$ free throws.

1. a) Calculate the relative frequency of scoring for each player.
2. b) Which of the two has a better success rate?
3. c) The coach has to choose one of them to take the decisive shot and says: "I trust Miriam's data more because it comes from many more shots." What does he mean? Do you agree with his reasoning? Argue using what you know about relative frequency.

A math app simulates rolling a die $10,000$ times and obtains these results: $1$ comes up $1,660$ times; $2$, $1,668$ times; $3$, $1,655$ times; $4$, $1,670$ times; $5$, $1,647$ times; and $6$ comes up $2,700$ times.

1. a) Calculate the relative frequency of each outcome, to four decimal places.
2. b) If the virtual die were fair, what relative frequency would you expect for each face, approximately?
3. c) Which face do you suspect is "loaded"? Explain your reasoning by comparing the obtained frequencies with the expected ones.