Sample Spaces and Events: Quick Revision

A brief note on Sample Spaces and Events. Read it before a test. Try each question at the end before you open its answer.

Sample space

• Sample space $S$: the set of all outcomes of a random experiment. $n(S)$ counts them.
• Steps multiply: three coin tosses give $2^3 = 8$ outcomes, and two dice give $6^2 = 36$ ordered pairs.
• Cards: $52$ cards, four suits of $13$, with $12$ face cards, $26$ red and $26$ black.
• Draws from $n$ different balls: $n^r$ with replacement, $^nP_r$ in order without replacement, $^nC_r$ together.
• Equally likely: use the ordered pairs for two dice, never the $21$ pairs without order.

Events

• Event: a subset of $S$. It happens when the outcome is in it.
• Types: simple (one outcome), compound (more), sure ($S$), impossible ($\varnothing$).
• Mutually exclusive: $A \cap B = \varnothing$. Exhaustive: $A \cup B = S$.
• Words: at least $k$ means $k$ or more. At most $k$ means $k$ or fewer. Exactly $k$ means only $k$.

The algebra of events

• Union, intersection, complement: $A \cup B$ is "$A$ or $B$", $A \cap B$ is "$A$ and $B$", $A'$ is "not $A$".
• Difference: $A - B = A \cap B'$, "$A$ but not $B$".
• De Morgan: $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$.
• Counting: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$.
• Three events: add the singles, take away the pairs, add the triple.

A worked example

In a class of $50$ students, $28$ play chess, $21$ play draughts and $9$ play both. Find the number who play neither.

1. At least one. $28 + 21 - 9 = 40$.
2. Neither. $50 - 40 = 10$.
3. Exactly one. $40 - 9 = 31$.

Practice with new numbers

A program to try

The program lists the outcomes of a draw of two tokens from a bag of four, in three ways, and then forms events with sets.

Mistakes to avoid

• Two dice as $21$ pairs. Use the $36$ ordered pairs.
• Forgetting the overlap in $n(A) + n(B)$.
• Mixing exclusive with independent. Exclusive events cannot happen together.
• Complementing a union wrongly. $(A \cup B)'$ is $A' \cap B'$.

Check yourself

1. How many outcomes does a sample space have when a coin is tossed twice and a die is thrown once?

2. A bag holds $5$ different balls. Two are taken out one after the other without replacement. How many ordered outcomes are there?

3. Are "the number is odd" and "the number is $6$" mutually exclusive for a die? Are they exhaustive?

4. If $n(A) = 14$, $n(B) = 10$ and $n(A \cup B) = 20$, find $n(A \cap B)$.

5. Write "exactly one of $A$ and $B$ happens" using the operations on sets.

Sources

• NCERT, Mathematics for Class XI (earlier edition), Probability, sections on the sample space, events and the algebra of events.
• OpenStax, Introductory Statistics, Chapter 3, sections on terminology and on independent and mutually exclusive events.