Classical Definition of Probability

What is the classical definition of probability? Probability is the measure of how likely a particular event is to happen. In Class 10 we study theoretical (classical) probability, which is defined only when all outcomes of a random experiment are equally likely:

$P(E)=\dfrac{\text{number of outcomes favourable to }E}{\text{number of all equally likely outcomes}}=\dfrac{n(E)}{n(S)}$

Here $S$ is the sample space (the set of all possible outcomes) and $E$ is the event (a subset of $S$). A single outcome of the experiment is called an elementary event; the probability of all the elementary events of an experiment always adds up to $1$.

Types of events you must know:

• Sure (certain) event — it is bound to happen, so $P(\text{sure})=1$. Example: getting a number less than $7$ on a die.
• Impossible event — it can never happen, so $P(\text{impossible})=0$. Example: getting $8$ on a single die.
• Complementary event — the event "$E$ does not happen", written $\overline{E}$ (or $E'$). For every event, $P(E)+P(\overline{E})=1$, so $P(\overline{E})=1-P(E)$.

Because the favourable outcomes can never exceed the total, the probability of any event always lies between $0$ and $1$:

$0\le P(E)\le 1$

The three standard experiments. Almost every board question is built from one of these sample spaces, so memorise them:

• Coins. One coin: $S=\{H,T\}$, $n(S)=2$. Two coins: $S=\{HH,HT,TH,TT\}$, $n(S)=4$. Three coins: $n(S)=8$. Each extra coin doubles the count ($2^n$).
• Dice. One die: $S=\{1,2,3,4,5,6\}$, $n(S)=6$. Two dice: $n(S)=36$ ordered pairs $(a,b)$ — see the sum table below.
• Cards. A well-shuffled deck has $52$ cards in $4$ suits — spades and clubs are black, hearts and diamonds are red ($26$ red, $26$ black). Each suit has $13$ cards: Ace, $2$ to $10$, Jack, Queen, King. There are $4$ aces, and the face cards are the Jack, Queen and King of each suit — $12$ face cards in all (not $16$: the Ace is not a face card).

Common mistakes to avoid:

• Counting only $4$ kings but forgetting that a deck has exactly $52$ cards — always use $52$ as the denominator for a single draw.
• Treating the Ace as a face card. Face cards are J, Q, K only ($12$ cards).
• For two coins, writing $S=\{HH,HT,TT\}$ and using $n(S)=3$. The order matters, so $HT$ and $TH$ are different — $n(S)=4$.
• Reading "at least one" as "exactly one". "At least one head" on two coins means $\{HH,HT,TH\}$, which is fastest as $1-P(\text{no head})=1-\tfrac14=\tfrac34$.
• Leaving the answer unsimplified — always reduce $\tfrac{8}{52}$ to $\tfrac{2}{13}$.

Probability is favourable-over-total, never anything else. Express it as a fraction in lowest terms, a decimal, or a percentage, but the value can never be negative and can never exceed $1$ — if your answer comes out as $\tfrac{7}{4}$ or $-\tfrac{1}{3}$, you have miscounted somewhere. A handy check: the probabilities of an event and its complement must add to exactly $1$.

The complement trick is the single most useful idea in this chapter: whenever an event is hard to count directly (especially "at least" problems), count the opposite and subtract from $1$. For instance, "at least one head in two tosses" is much faster as $1-P(TT)=1-\tfrac14=\tfrac34$ than by listing all the favourable cases.