Week 10 · Random experiments and sample spaces: coin and dice tables, tree diagrams, drawing with and without replacement, and events · 3 lessons
1
Random experiments: when an experiment is random, and the sample space S with its size n(S) for one, two and three coins
Statistics Sec 3 Term 1, Unit 3, Lesson 1
GoalSay when an experiment is random, and write the sample space S and its size n(S) for one, two and three coin tosses.
New words
random experiment · all possible outcomes are known before it is done, but we cannot tell which one will happensample space S · the set of all possible outcomes of the experimentn(S) · the number of elements of the sample space
Explanation
Probability begins with an experiment whose result we cannot predict. We call it random when two things are true: before we do it we know every possible outcome, and when we do it we cannot say which of them will happen. Tossing a coin is random: the outcomes are head (H) or tail (T), and we cannot tell which comes. The set of all possible outcomes is the sample space S, and n(S) is how many elements it has. One toss gives S = {H, T} and n(S) = 2. Two tosses give pairs written in order: (H, H), (H, T), (T, H), (T, T), so n(S) = 4 = 2². The pair (H, T) (head first, tail second) is not the same outcome as (T, H). Tossing two different coins together gives the same sample space as tossing one coin twice. With m tosses, n(S) = 2^m. The tree shows two tosses, one branch for each outcome.
Worked examples
Which are random experiments? (a) Rolling a die and noting the top number. (b) Drawing one letter card from a bag that holds the cards A, B, C and D, and noting the letter. (c) Drawing a card from a bag whose cards we have never seen. (a) and (b) are random: all outcomes are known beforehand and we cannot predict which will happen. (c) is not random in this book’s sense: we do not know the possible outcomes beforehand.
Two coins: the sample space. S = {(H, H), (H, T), (T, H), (T, T)}. Reading the tree from the left: first the outcome of the first toss, then the outcome of the second. The outcome (T, H) means tail on the first, head on the second. n(S) = 2 × 2 = 4.
Three tosses. Each of the 4 outcomes of two tosses has two continuations, so n(S) = 2 × 2 × 2 = 8: S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. The tree has 8 end points, one for each of these outcomes.
Class activity
Toss a coin 3 times and write the outcome as a triple, for example HTH. Do this 8 times. Did you see all 8 outcomes of S? Why is it not surprising if some are missing and some appear twice?
Practice
1
Which is a random experiment: (a) tossing a coin, (b) adding 2 and 3, (c) spinning a spinner with three equal coloured parts and noting the colour?
(a) and (c). Adding 2 and 3 always gives 5, so there is nothing to predict.
2
Write the sample space for tossing two coins and say n(S).
S = {(H, H), (H, T), (T, H), (T, T)}, n(S) = 4.
3
Find n(S) for 4 tosses and for 5 tosses.
2⁴ = 16 and 2⁵ = 32.
4
Why are (H, T) and (T, H) two different outcomes?
The order matters: in the first the first toss is a head, in the second it is a tail.
5
Two different coins (one small, one large) are tossed together. How many outcomes are in S?
4, the same as for one coin tossed twice: (small, large) = (H, H), (H, T), (T, H), (T, T).
