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Probability in society and science

Science and society, grade 10 (3 units) Lesson 11 of 15Learning

Probability in society and science

Choosing at random from a population

When a person is chosen at random from a population, the probability that they have some property is the share of the population that has it.

“At random” means that every person has the same chance of being chosen. Then each person is an equally likely outcome, and:

P(A)=kn\displaystyle P(A)=\frac{k}{n}

k\displaystyle k – the number of people with the property, n\displaystyle n – the number of people in the population.

In a town of 50,000 residents, the blood groups are spread like this:

Blood groupOABAB
Residents19,00018,0009,0004,000

A resident is chosen at random. The probability that their blood group is A:

1800050000=0.36\frac{18000}{50000}=0.36
A fraction, a decimal and a percent

The same probability is written in three ways: 36100=0.36=36%\displaystyle {\frac{36}{100}}={0.36}={36\%}.

A medicine’s leaflet sometimes says “1 in 10,000” – a very small probability:

110000=0.0001=0.01%\frac{1}{10000}=0.0001=0.01\%
  • “Common”: up to 1 in 10 users, that is up to 10%.
  • “Rare”: up to 1 in 1,000 users, that is up to 0.1%.
The complement

The probability that an event does not happen is 1 minus the probability that it does:

P(A‾)\displaystyle P(\overline{A}) =\displaystyle = 1−P(A)\displaystyle 1-P(A)

In the town of the example, the probability that a resident chosen at random has blood group O is 1900050000=0.38\displaystyle {\frac{19000}{50000}}={0.38}. So the probability that their group is not O:

1−0.38=0.621-0.38=0.62
“At least” by the complement

“At least” takes in several values. It is often easier to work out the complement and take it from 1.

The number of children in the families of one town:

Children01234 or more
Families20%25%30%15%10%

A family is chosen at random.

  • At least one child – the complement: no children. 1−0.2=0.8\displaystyle {1-0.2}={0.8}
  • At least 2 children – the complement: 0 or 1 children. 1−(0.2+0.25)=0.55\displaystyle {1-(0.2+0.25)}={0.55}

Check: 0.3+0.15+0.1=0.55\displaystyle {0.3+0.15+0.1}={0.55}.

A raffle

In a raffle, each ticket is an equally likely outcome. The probability of winning is the number of tickets you bought divided by the number of tickets sold.

A raffle for the school trip sold 2,000 tickets at 10 ₪ each.
The prize: an electric bicycle worth 5,000 ₪.

  • One ticket: 12000=0.0005=0.05%\displaystyle {\frac{1}{2000}}={0.0005}={0.05\%}.
  • 20 tickets, for 200 ₪: 202000=0.01=1%\displaystyle {\frac{20}{2000}}={0.01}={1\%}.
  • The organisers took in 2000⋅10=20000\displaystyle {2000\cdot10}={20000} ₪ and gave a prize of 5,000 ₪: for every 10 ₪ paid for a ticket, the buyers get back 50002000=2.5\displaystyle {\frac{5000}{2000}}={2.5} ₪ on average.

Buying a ticket is a gift to the trip. As an investment, the ticket does not pay.

Exercise 1

On a blood donation day, 200 people gave blood.
Their blood groups:

Blood groupOABAB
Donors70664024

A unit of blood is chosen at random. What is the probability that its group is:

a

A?

    / – 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

    not O?

      / – 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

      B or AB?

        / – 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 2

        A medicine’s leaflet says: “Dizziness – in 3 of every 200 users”.

        a

        Write the probability that a user feels dizzy as a decimal.

          / – 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

          Write it as a percent.

            / – 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

            The number of cars in the households of one village:

            Cars0123 or more
            Households15%45%30%10%

            A household is chosen at random. What is the probability that it has:

            a

            at least one car?

              / – 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

              at least two cars?

                / – 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

                Each of two raffles has one prize.
                Raffle A sold 300 tickets, and Dan bought 3 of them.
                Raffle B sold 800 tickets, and Dan bought 6 of them.

                a

                What is the probability that Dan wins raffle A?

                  / – 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

                  What is the probability that Dan wins raffle B?

                    / – 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

                    In which raffle is Dan’s chance of winning greater? Explain.