Relative frequency and experimental probability
You toss a coin 4 times and get heads 3 times. Does that mean the probability of heads is $0{,}75$? No: with only a few tosses, chance can surprise you. If you repeat it 1000 times, the result gets much closer to a half.
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 $4$ colours 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 $4$ 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?
Relative frequency settles down as you repeat
Roll the die many times and watch how the bars (what really 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?
Few repetitions can be misleading
With few attempts, the relative frequency can stray a long way from the real probability, even though the latter 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?
When you cannot calculate with Laplace
Sometimes we do not know whether the outcomes are equally likely (a drawing pin, a knucklebone, the lid of a jar as it lands). In those cases, we 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?