Probability Class 11: notes and important questions

Probability: revision notes

1. Random experiments and sample spaces

A random experiment has more than one possible outcome and the outcome cannot be predicted. The sample space \(S\) is the set of all outcomes: three coins give \(2^3 = 8\) outcomes; two dice give \(36\) ordered pairs.

2. Events

• An event is a subset of \(S\). Impossible event \(\phi\); sure event \(S\); simple event: one outcome.
• 'Not \(A\)' is \(A' = S - A\); '\(A\) or \(B\)' is \(A \cup B\); '\(A\) and \(B\)' is \(A \cap B\); '\(A\) but not \(B\)' is \(A - B = A \cap B'\).
• Mutually exclusive: \(A \cap B = \phi\). Exhaustive: \(A_1 \cup A_2 \cup \cdots = S\).

3. Axiomatic probability

A probability \(P\) assigns to each event a number with (i) \(P(E) \ge 0\), (ii) \(P(S) = 1\), (iii) \(P(E \cup F) = P(E) + P(F)\) when \(E, F\) are mutually exclusive. For a finite sample space the probabilities of the outcomes are non-negative and add to \(1\). With \(n\) equally likely outcomes, \(P(E) = \dfrac{n(E)}{n(S)}\) (the classical definition).

4. Rules

• \(P(\text{not } A) = 1 - P(A)\); \(0 \le P(A) \le 1\); \(P(\phi) = 0\).
• \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\); for mutually exclusive events \(P(A \cup B) = P(A) + P(B)\).
• \(P(A \cap B') = P(A) - P(A \cap B)\); \(P(\text{neither}) = 1 - P(A \cup B)\).

5. Counting with combinations

When a selection is made at random, count favourable and total selections with \({}^nC_r\): for example the probability that a random committee has exactly \(2\) girls is \(\dfrac{{}^gC_2 \times {}^bC_{r-2}}{{}^{g+b}C_r}\).

Worked example 1

Two dice are thrown. Find \(P(\text{sum} = 5)\).

\((1,4), (2,3), (3,2), (4,1)\): \(\dfrac{4}{36} = \dfrac19\).

Worked example 2

\(P(A) = 0.6\), \(P(B) = 0.5\), \(P(A \cup B) = 0.8\). Find \(P(A \cap B)\).

\(0.6 + 0.5 - 0.8 = 0.3\).

Worked example 3

Two cards are drawn from \(5\) cards numbered \(1\) to \(5\). Find the probability that both are odd.

\(\dfrac{{}^3C_2}{{}^5C_2} = \dfrac{3}{10}\).

Common errors

• Adding \(P(A) + P(B)\) for events that can happen together (the overlap is counted twice).
• Listing unordered outcomes for two dice, which are not equally likely.
• Assigning probabilities that are negative or do not add up to \(1\).
• Confusing "mutually exclusive" (cannot both happen) with "exhaustive" (between them cover everything).

Exam tips

• Name the events with letters and write the rule you use before substituting.
• Check that every answer lies between \(0\) and \(1\).