Probability

Probability is a numerical measure of the chance that a particular event will occur. When we toss a coin, roll a die or draw a card, we cannot say for sure what the result will be, but we can measure how likely each result is.

In earlier classes you studied experimental (empirical) probability, which is found by actually performing an experiment many times. In Class 10 we focus on theoretical probability, also called classical probability. Here we do not perform experiments; instead we reason about the outcomes, assuming all of them are equally likely.

An outcome is a single result of an experiment. The set of all possible outcomes is the sample space. Outcomes are said to be equally likely if none of them has a greater chance of occurring than another. For a fair coin, Head and Tail are equally likely; for a fair die, each of the six faces is equally likely.

Worked example: A fair coin is tossed once. The possible outcomes are Head (H) and Tail (T). There are 2 equally likely outcomes. The chance of getting a Head is 1 out of 2, so its probability is 1/2. Similarly the probability of a Tail is 1/2.

An event is a collection of one or more outcomes of an experiment. For example, when a die is thrown, 'getting an even number' is an event made up of the outcomes 2, 4 and 6.

If all outcomes are equally likely, the probability of an event E is given by:

P(E) = (number of outcomes favourable to E) / (total number of possible outcomes)

An event with exactly one outcome is called an elementary event. For a single die throw, 'getting a 5' is an elementary event. A key fact is that the sum of the probabilities of all the elementary events of an experiment is 1.

Worked example: A die is thrown once. Find the probability of getting an even number.

• Total outcomes = 6 (the numbers 1, 2, 3, 4, 5, 6).
• Favourable outcomes (even numbers) = 2, 4, 6, that is 3 outcomes.
• P(even number) = 3/6 = 1/2.

Second example: A bag contains 4 red and 6 green balls of the same size. One ball is drawn at random. Total outcomes = 10, favourable (red) = 4, so P(red) = 4/10 = 2/5.

Because the number of favourable outcomes can never be less than 0 nor more than the total number of outcomes, the probability of any event always satisfies:

0 ≤ P(E) ≤ 1

An impossible event is one that cannot happen at all. It has no favourable outcome, so its probability is 0. For example, getting the number 7 on an ordinary die is impossible, so its probability is 0/6 = 0.

A sure event (also called a certain event) is one that must happen. Every outcome is favourable, so its probability is 1. For example, getting a number less than 7 on an ordinary die is certain, so its probability is 6/6 = 1.

Worked example: A die is thrown once. Find the probability of getting a number less than 7 and the probability of getting the number 8.

• Numbers less than 7 on a die: 1, 2, 3, 4, 5, 6, that is all 6 outcomes. P = 6/6 = 1 (a sure event).
• The number 8 does not appear on a die: 0 favourable outcomes. P = 0/6 = 0 (an impossible event).

Any value that is negative or greater than 1 can never be a probability. So values such as −0.4 or 1.5 or 7/3 are not valid probabilities.

For an event E, the event that E does not happen is called the complement of E, written as Ē (E-bar) or 'not E'. Together, E and 'not E' cover every possible outcome, and they have no outcome in common.

Because one of E or 'not E' is bound to occur, their probabilities add up to 1:

P(E) + P(not E) = 1, which rearranges to P(not E) = 1 − P(E).

This is very useful: when the favourable outcomes for 'not E' are easier to count, we find P(not E) first and subtract.

Worked example: The probability that it will rain tomorrow is 0.85. What is the probability that it will not rain?

• P(not rain) = 1 − P(rain) = 1 − 0.85 = 0.15.

Second example: A die is thrown once. Let E be the event 'getting a number greater than 4'. The favourable outcomes are 5 and 6, so P(E) = 2/6 = 1/3. Then the probability of getting a number 4 or less is P(not E) = 1 − 1/3 = 2/3. We can check directly: numbers 4 or less are 1, 2, 3, 4, giving 4/6 = 2/3.

Most board questions come from three familiar experiments. Learning their sample spaces makes the counting easy.

Coins. One coin has 2 outcomes: H, T. Two coins (or one coin tossed twice) have 4 equally likely outcomes: HH, HT, TH, TT.

Dice. One die has 6 outcomes: 1, 2, 3, 4, 5, 6. Two dice thrown together have 6 × 6 = 36 equally likely outcomes, written as ordered pairs like (1,1), (1,2), ... , (6,6).

Playing cards. A standard deck has 52 cards in 4 suits of 13 each. The suits hearts and diamonds are red (26 cards); spades and clubs are black (26 cards). Each suit has an Ace, numbers 2 to 10, and three face cards: Jack, Queen, King. So there are 12 face cards in all (4 Jacks, 4 Queens, 4 Kings) and 4 Aces. Note: Aces are not counted as face cards.

Worked example (two coins): Two coins are tossed together. Find the probability of getting exactly one head. Sample space = {HH, HT, TH, TT}, total 4. Exactly one head: HT and TH, that is 2 outcomes. P = 2/4 = 1/2.

Worked example (cards): One card is drawn from a well-shuffled deck of 52 cards. Find the probability that it is a red face card. Red face cards = 2 red suits × 3 face cards each = 6. P = 6/52 = 3/26.