Question

Experiment: Tossing two coins. Random variable 𝑋 = number of heads a) List all possible sample space. b) Find the random variable values. c) Find the probabilities for the random variable values.

Answer 1

a) Let's use the letters H and T for “head” and “tail” respectively. The sample space:

$S = \begin{Bmatrix} TT, HT, TH, HH\end{Bmatrix}.$

b) In the case of TT (i.e we got two tails) X = 0; for HT and TH (first head, second tail or vice versa) X = 1 and for HH (both heads) X = 2.

c) The probability that we get head $p = \frac{1}{2},$ that we get tail $q = 1-p=1-\frac{1}{2}=\frac{1}{2}.$

The result of both tossings are independent events, for each sample point:

$P(TT)=P(T)\cdot P(T)=q\cdot q =\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{4};\\ P(TH)=P(T)\cdot P(H)=q\cdot p =\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{4};\\ P(HT)=P(T)\cdot P(H)=p\cdot q =\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{4};\\ P(HH)=P(T)\cdot P(H)=p\cdot p =\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{4}.$

The probabilities for the random variable values:

$P(X=0)=P(TT)=\frac{1}{4};\\ P(X=1)=P(TH)+P(HT)=\frac{1}{4}+\frac{1}{4}=\frac{1}{2};\\ P(X=2)=P(HH)=\frac{1}{4}.$