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← The Mathematics of Maybe: Introduction to Probability

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Lesson 10 of 11

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Eight ideas, eight checks. The table comes first, so you have somewhere to look.

Eight ways to be exact about something you cannot call in advance
IdeaHow it goes
RandomThe next toss cannot be called. A whole run can be described.
The 0-to-1 scaleA chance is a number from 0 to 1. 0 is impossible. 1 is certain. The chances of all the outcomes of one draw add to 1.
Observed chanceAn observed chance is a count over a whole. It is the outcome's trials over every trial made.
SettlingMore tries steady the share. They never fully still it. Each try is blind to the ones before it.
Worked-out chanceCount the outcomes an event allows. Divide by all the outcomes that can happen. First check they are equally likely.
Two figuresReal rolls give one figure. The die's faces give another. With few rolls a gap is normal. It does not prove the die is unfair.
PairsWrite the two actions as a table of pairs. Count the pairs an event holds. Divide by all the pairs.
TreesOne stage per choice. Multiply along a path. The paths together add to exactly 1.

A friend calls heads on every toss of a fair coin. What happens over a long run?

  • Heads come up more, since she keeps calling heads
  • Tails come up more, since tails must balance her calls
  • About half are right, since any plan scores half

A jar has 5 red, 3 blue and 2 green marbles. One is drawn. What do the three chances add to?

  • Exactly 1, since 0.5 + 0.3 + 0.2 = 1
  • 1/3, since three colours share evenly
  • 10, since 5 + 3 + 2 = 10

Twenty tosses gave 12 heads. What is the observed chance of tails?

  • 12/8, since heads is compared with tails
  • 8/20 = 0.4, since tails is over every toss
  • 12/20 = 0.6, since that is the count of heads

A fair coin has landed tails four times in a row. What does that tell you about the next toss?

  • A head is due, since tails is on a run
  • A tail is more likely, since tails is on a run
  • Nothing, since each toss ignores the last

A jar has 5 red, 3 blue and 2 green marbles. Someone says each colour has a chance of 1/3. What is wrong?

  • Not equally likely, since the colours hold 5, 3 and 2 marbles
  • Nothing is wrong, since three colours always share evenly
  • Each chance is 1/10, since ten marbles are in the jar

A die is rolled 15 times and shows a 4 five times. Experimental: about 0.33. Theoretical: about 0.17. What does the gap say?

  • The count is wrong, since the figures must match
  • It says little, since few rolls can stray this far
  • The die is unfair, since 0.33 is well above 0.17

Two coins are tossed together. What is the chance that exactly one lands heads?

  • 1/3, since results are two heads, one head, or none
  • 1/4, since only one pair of four fits
  • 1/2, since HT and TH hold exactly one head

On the two-coin tree, what is the chance of heads then tails? What do the four paths add to?

  • 1/4, and the paths add to exactly 1
  • 1/2, and the paths add to 1/2 in total
  • 1/3, and the paths add to 1 in total

Chapter 7, The Mathematics of Maybe, is next. Read it and see how each idea in the table grows.

Next: Predicting What Comes Next: Exploring Sequences and Progressions. From chances to rules that name every term.

Reveal every answer to finish this lesson.