Answer
a)The events B and C are said to be mutually exclusive.
b)The set B is subset of A.
Work Step by Step
(a)
A coin is to be tossed four times, so there are 16 possible outcomes in sample space S:
\[S=\left\{ \begin{array}{*{35}{l}}
HHHH,\text{ }HHHT,\text{ }HHTH,\text{ }HTHH, \\
THHH,\text{ }HHTT,\text{ }HTHT,\text{ }THHT, \\
THTH,\text{ }TTHH,\text{ }HTTT,\text{ }THTT, \\
TTHT,TTTH,\text{ }HTTH,\text{ }TTTT \\
\end{array} \right\}\]
Define events A, B and C such that
A: exactly two heads appear
B: heads and tails alternate
C: first two tosses are heads
Hence, the possible outcomes of events A, B and C are:
$A=\{HHTT,\text{ }HTHT,\text{ }THTH,\text{ }THHT,\text{ }HTTH,\text{ }TTHH\}$
$B=\{HTHT,\text{ }THTH\}$
$C=\{HHHH,\text{ }HHHT,\text{ }HHTH,\text{ }HHTT\}$
We know that events A and B defined over the same sample space are said to be mutually exclusive if they have no outcomes in common; that is \[(A\cap B)=\varnothing \]
Since, from above events we can see that event B and C have no outcomes in common: \[(B\cap C)=\varnothing \]
Therefore, events B and C are said to be mutually exclusive.
(b)
The possible outcomes of events A, B and C are:
$A=\{HHTT,\text{ }HTHT,\text{ }THTH,\text{ }THHT,\text{ }HTTH,\text{ }TTHH\}$
$B=\{HTHT,\text{ }THTH\}$
$C=\{HHHH,\text{ }HHHT,\text{ }HHTH,\text{ }HHTT\}$
We know that set A is a subset of another set B if all elements of set A are elements of set B.
From above events we can see that all outcomes of event B are outcomes of event A.
Therefore, event B is subset of A.
