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Relative frequency as an estimate of probability

Science and society, grade 10 (3 units) Lesson 13 of 15Learning

Relative frequency as an estimate of probability

Estimating a probability from a lot of data

When the outcomes are not equally likely, a probability is estimated by the relative frequency: in how many of many times the event happened.

P(A)≈fn\displaystyle P(A)\approx\frac{f}{n}

f\displaystyle f – the number of times the event A\displaystyle A happened, n\displaystyle n – the number of times it was checked.

In one country, 180,000 babies were born in one year, and 92,340 of them were boys:

92340180000=0.513\frac{92340}{180000}=0.513

The probability that a newborn baby is a boy is about 0.513 – a little more than a half. A birth is not the toss of a coin: symmetry alone would get it wrong.

More data, a better estimate

The more data there are, the closer the relative frequency comes to the probability.

A drawing pin thrown on a table lands on its back (point up) or on its side.
A computer simulated 1,000 throws of a pin whose probability of landing on its back is 0.6 – the dashed line:

  • After a few throws, the relative frequency jumps up and down.
  • After hundreds of throws, it settles near 0.6.
  • A real pin does not come with its probability: it is thrown many times, and the probability is estimated by the relative frequency.
Insurance: an estimate from past data

An insurance company estimates a probability from past data, and works out from it how many times the event is expected to happen:

n⋅P(A)\displaystyle n\cdot P(A)

Last year the company insured 40,000 young drivers, and 2,800 of them had an accident and made a claim.
Next year it will insure 50,000 young drivers.
A claim costs the company 12,000 ₪ on average.

  • The estimated probability of an accident in a year: 280040000=0.07\displaystyle {\frac{2800}{40000}}={0.07}.
  • The expected number of claims next year: 50000⋅0.07=3500\displaystyle {50000\cdot0.07}={3500}.
  • The expected cost: 3500⋅12000=42000000\displaystyle {3500\cdot12000}={42000000} ₪.
  • For each driver: 4200000050000=840\displaystyle {\frac{42000000}{50000}}={840} ₪. That is the lowest price the company can charge without losing money.
“70% rain” in a forecast

“A 70% chance of rain tomorrow” means: of the days when the forecast said 70%, it rained on about 70%.

  • It does not mean that rain will fall on 70% of the area, or for 70% of the hours.
  • A single forecast cannot be checked: tomorrow it rains or it does not.
  • Many forecasts are checked together: if it rained on 69 of 100 days with a 70% forecast, the forecast is good.
When the estimate is good

An estimate from relative frequency is good only when the data fit the question.

  • Enough data: a player who scored 3 of 3 shots does not score with probability 1.
  • The same population: data on young drivers do not fit an estimate of the accidents of 50-year-old drivers.
  • The same conditions: the rain data of an especially wet year do not describe an ordinary year.
Exercise 1

A computer simulates a basketball player taking free throws.
His probability of scoring is not written here: run the experiment to estimate it.

  1. Run 10 throws, and write the relative frequency of the baskets.
  2. Add 100 and then 1,000 throws, and write the relative frequency again.
  3. What is your estimate of the probability of scoring? Which of the estimates is better, and why?
Exercise 2

3,600 babies were born in a hospital in one year, and 1,836 of them were boys.

a

Estimate the probability that a baby born in the hospital is a boy.

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b

    Next year the hospital expects 5,000 births. How many boys are expected to be born?

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      Exercise 3

      A company insures phones against a broken screen.
      Last year it insured 25,000 phones, and 1,500 of them broke.
      Next year it will insure 30,000 phones.
      A screen repair costs the company 600 ₪.

      a

      Estimate the probability that an insured phone breaks in a year.

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        b

        How many phones are expected to break next year?

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          c

          What is the least the company must charge for each phone to cover the expected repairs? (in shekels)

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            Exercise 4

            A weather service forecast “30% rain” on 200 days.
            It rained on 58 of those days.

            a

            What is the relative frequency of rainy days among the 200 days?

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              b

              Are the “30% rain” forecasts good? Explain.

              Exercise 5

              Which estimate is better? Explain each answer.

              1. Noa scored 7 of 10 shots, and Maya 140 of 200. Which of them can estimate her probability of scoring better?
              2. An insurance company wants to estimate the probability of an accident for 60-year-old drivers. Should it use the data of drivers aged 18–24?