14. Probability

8 topics2 h

Chapter 14 of NCERT Mathematics for Class 10: Probability. Three sections — From counting trials to reasoning about them, What the number can be and Counting the outcomes correctly. Eight videos, 2 h in all.

Worked answers to the 25 exercise questions in this chapter: Exercise 14.1

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Part 1

From counting trials to reasoning about them

12 minEqually likely outcomes, and the everyday cases where that failsA coin has two results and the answer is a half. A bag of four red balls and one blue one also has two results - red or blue - and the answer is four fifths. So counting the results you have named decides nothing. Equal likelihood is not a property of the experiment; it is a property of the way you carved the experiment into results, and you have to earn it before you divide.12 min · 12 parts11 minWhy repeating an experiment is often impossible, and what replaces itA satellite launch is too costly to repeat and an earthquake cannot be staged at all, so the ordinary definition of probability — run the experiment, count, divide — has nothing left to count.11 min · 12 parts12 minFavourable over total: the definition this chapter runs onAcross 510 events from every experiment up to eight outcomes, favourable over unfavourable gives the right answer 8 times, and two ways so a half gives it 98 times — under one in five.12 min · 12 parts

Part 2

What the number can be

11 minSingle-outcome events, and why all of them together come to 1Three blue marbles, two white, four red: none of the three colour events is elementary, yet their probabilities still add to exactly 1 — no proof that the pieces were equal, or even that they form a partition.11 min · 11 parts11 minThe impossible and the certain pin the scale at 0 and at 1Ask a die for an 8 and every outcome gets crossed off — nought over six, not a comment on luck but a count with nothing in it. Ask for a result below 7 and nothing is crossed off: six over six.11 min · 11 parts13 minComplements: knowing one probability hands you the otherA tennis match gives one player 0.62 and the other 0.38 by subtracting from 1 — quietly assuming no draw. Add one back in, at 0.62, 0.05 and 0.33, and the subtraction overstates the second player by exactly the draw.13 min · 12 parts

Part 3

Counting the outcomes correctly

16 minCoins, dice, bags and a deck of 52: getting the denominator rightTwo coins do not give three results. They give four — and merging heads-then-tails with tails-then-heads is the mistake this whole topic is written to catch.16 min · 14 parts11 minWhen outcomes cannot be counted, measuring length or area insteadStop a two-minute piece of music at a random instant, and the result is a time — one of infinitely many. Favourable over total turns into infinity over infinity, and the ratio simply jams.11 min · 11 parts

What you will be able to do

  • State what is being assumed when a coin is called fair, and separate that from what is being assumed when the toss is called random
  • Write down the Class IX formula in terms of what each of its two counts measures, and say why both counts require the experiment to have been performed
  • State the definition in terms of what each of its two counts counts, and say which clause of it carries the assumption
  • Define an elementary event and decide, for any event in the chapter, whether it is one
  • Compute the probability of an event no outcome favours, and name what the chapter calls such an event
  • Recognise, in a pair of worked answers, that the second event is the denial of the first
  • Fix the denominator for a described experiment by listing or counting outcomes that are interchangeable, before computing anything
  • Explain why the counting definition cannot be applied as it stands when the outcomes fill an interval or a region

What this chapter assumes you already know

  • Probability from Class IX as a fraction obtained by running an experiment many times and tallying how often something happened
  • Reading a fraction as a part of a whole, and simplifying it
  • The idea that a physical object can be symmetric — that a shape can have two sides with nothing to tell them apart
  • Listing every possibility in a small situation without missing one or writing one twice
  • The Class IX treatment of probability as a fraction obtained by tallying repeated trials
  • Reading a fraction as a proportion, and comparing two fractions
  • The idea that a fraction obtained from a small sample can move about, while one obtained from a large sample tends to settle
  • Listing every possibility in a small experiment, exhaustively and without duplication

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "Two possible results, so each has probability one-half."
  • "Fair means the person tossing is honest."
  • "Equally likely is a property of the experiment."
  • "If the outcomes are not equally likely, nothing can be computed."
  • "A coin could land on its edge, so the model is wrong."
  • "The chapter proved that dice are fair."
  • "The theoretical value is the true one and the experimental value is a bad copy of it."
  • "You can always just do the experiment more times."