Week 10 · Random experiments and sample spaces: coin and dice tables, tree diagrams, drawing with and without replacement, and events · 3 lessons
3
An event is a subset of S: simple, sure and impossible events, and writing events such as at least, at most and exactly
Statistics Sec 3 Term 1, Unit 3, Lesson 1
GoalSay that an event is a subset of S, name simple, sure and impossible events, and write events such as at least, at most and exactly.
New words
event · a subset of the sample space Ssimple, sure, impossible event · one element; S itself; the empty set â at least, at most, exactly · âat least 2â means 2 or more; âat most 2â means 2 or fewer; âexactly 2â means 2 only
Explanation
An event is something we may ask about the result of the experiment, and we write it as the set of the outcomes that make it happen: so an event is a subset of S. An event with one element is a simple event; the event S itself is the sure event (it always happens); the empty set â is the impossible event (it never happens). In words, âat least kâ means k or more, âat most kâ means k or fewer, and âexactly kâ means k and nothing else. To write an event, go through S and keep the outcomes that satisfy the words. For three coin tosses, S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. The event âat least two headsâ keeps HHH, HHT, HTH and THH. For two dice, the event âthe sum is 9â keeps the pairs shaded in the table.
1st \ 2nd
1
2
3
4
5
6
1
(1, 1)
(1, 2)
(1, 3)
(1, 4)
(1, 5)
(1, 6)
2
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
3
(3, 1)
(3, 2)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
4
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
5
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
6
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)
Worked examples
One die: S = {1, 2, 3, 4, 5, 6}. A âan even numberâ = {2, 4, 6}. B âa number greater than 4â = {5, 6}. C âthe number 3â = {3} is a simple event. D âa number less than 7â = S is the sure event. E âthe number 7â has no outcome, so E = â is the impossible event.
Three coin tosses. âAt least two headsâ = {HHH, HHT, HTH, THH}, 4 outcomes. âAt most one headâ = {HTT, THT, TTH, TTT}, 4 outcomes. âExactly two tailsâ = {HTT, THT, TTH}, 3 outcomes. âExactly three headsâ = {HHH} is simple. âFour headsâ = â .
A stopping rule. A coin is tossed until a head appears or two tails have appeared, and then we stop. The outcomes are H, TH and TT, so S = {H, TH, TT}. The event âat most one tailâ = {H, TH} and âexactly two tailsâ = {TT}. Note: these three outcomes are not equally likely (a head on the first toss happens about half of the time, TT only about a quarter of the time), so here we only list the events and never count elements to find a probability.
Class activity
Roll two dice in your mind and write the pairs of the event âthe difference between the numbers is 3â. How many pairs does it have? Shade them on the 36-cell table.
Practice
1
One die. Write the event âa prime numberâ and the event âa number divisible by 3â.
{2, 3, 5} and {3, 6}.
2
Three coin tosses: write the event âexactly one headâ and say whether it is simple.
{HTT, THT, TTH}. It has three elements, so it is not simple.
3
Three coin tosses: write the event âat most two headsâ. Which outcome is missing from S?
All outcomes except HHH: 7 outcomes. HHH (three heads) is the only one missing.
4
Two dice: write the event âthe sum is 11â and the event âthe sum is 1â. Which kind of event is the second?
{(5, 6), (6, 5)} and â . The second is the impossible event (the smallest sum is 2).
5
Say which is simple, sure or impossible for one die: âa number less than 1â, âthe number 6â, âa whole number from 1 to 6â.
âLess than 1â is impossible (â ); âthe number 6â is simple ({6}); âa whole number from 1 to 6â is sure (S).
