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Experiments in several stages

Science and society, grade 11 (3 units) Lesson 12 of 16Learning

Experiments in several stages

A tree diagram of three stages

In an experiment of three stages, each branch of the second stage goes on into the third. Here too a path’s probability is the product of the probabilities on its branches:

P=p1⋅p2⋅p3\displaystyle P=p_1\cdot p_2\cdot p_3

A family has three children.
Suppose that at each birth a boy and a girl are equally likely, 12\displaystyle \frac{1}{2} each.
The path “three girls” is marked in the tree:

  • There are 2⋅2⋅2=8\displaystyle {2\cdot2\cdot2}={8} paths, each with probability 12⋅12⋅12=18\displaystyle {\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}}={\frac{1}{8}}.
  • Three girls: one path, 18\displaystyle \frac{1}{8}.
  • Exactly two boys: three paths – boy-boy-girl, boy-girl-boy, girl-boy-boy. Add them: 38\displaystyle \frac{3}{8}.
Drawing without putting back

When nothing is put back, each stage changes what is left: the next stage has one item fewer, and the probabilities on the branches change.

The class committee has 10 pupils: 6 girls and 4 boys.
Three of them are drawn, one after another, for the trip committee. Whoever has been drawn is not drawn again.
The path “three girls” is marked in the tree:

  • In the first draw there are 6 girls out of 10.
  • If a girl was drawn, 5 girls out of 9 are left; after another girl – 4 out of 8.
610⋅59⋅48=120720=16\frac{6}{10}\cdot\frac{5}{9}\cdot\frac{4}{8}=\frac{120}{720}=\frac{1}{6}
With and without putting back

When what was drawn is put back, every stage starts from the same state, and the probabilities are the same in all the stages.

Suppose the committee drew three times from all ten – and the same pupil could be drawn again:

610⋅610⋅610=2161000=0.216\frac{6}{10}\cdot\frac{6}{10}\cdot\frac{6}{10}=\frac{216}{1000}=0.216
  • Without putting back: 16≈0.167\displaystyle \frac{1}{6}\approx0.167 – less: each girl drawn leaves fewer girls behind.
  • When choosing from a large population – thousands of people – the difference is negligible, and the work is done as if with putting back.
At least once

“At least once” is the complement of “never”. Instead of adding many paths, work out one path and subtract it from 1.

On every flight of an airline, the probability of a delay is 0.2\displaystyle 0.2.
Noa flies 4 times, and delays on different flights do not depend on each other.
What is the probability that at least one of her flights is delayed?

  • No flight is delayed: 0.8⋅0.8⋅0.8⋅0.8=0.84=0.4096\displaystyle {0.8\cdot0.8\cdot0.8\cdot0.8}={0.8^{4}}={0.4096}.
  • At least one flight is delayed: 1−0.4096=0.5904\displaystyle {1-0.4096}={0.5904}.

In general, when the probability in each experiment is p\displaystyle p and n\displaystyle n independent experiments are made:

1−(1−p)n\displaystyle 1-(1-p)^{n}
Exercise 1

There are three traffic lights on the way to work.
At each light the probability of arriving while it is green is 25\displaystyle \frac{2}{5}, whatever the other lights do.

Write the probabilities on the branches and each path’s probability.

a

What is the probability that all three lights are green?

    / – 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 at least one light is red?

      / – 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 box holds 10 bulbs, and 2 of them are faulty.
      A quality inspector takes out 3 bulbs, one after another, and does not put them back.

      Write the probabilities on the branches and each path’s probability.

      a

      What is the probability that all three bulbs work?

        / – 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 the inspector finds at least one faulty bulb?

          / – 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

          In a multiple-choice test each question has four answers, and only one of them is right.
          Yoav has not studied, and he guesses the answers to three questions.

          Write the probabilities on the branches and each path’s probability.
          Mark the paths of the event “at least two right answers”, and find its probability.

            / – 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

            The probability that a tomato seed sprouts is 0.7\displaystyle 0.7.
            Shira plants 4 seeds, and each seed sprouts whatever the others do.

            What is the probability that at least one seed sprouts?

              / – 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 5

              A class has 30 pupils: 15 girls and 15 boys.
              The teacher draws two names.

              a

              What is the probability that both names are girls’, if the first name is put back after the draw (and the same girl can be drawn twice)?

                / – 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 both names are girls’, if the first name is not put back?

                  / – 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

                  Which probability is greater? Explain why.