Q. 1 (20 pts) (MT1-2007)
Assume two four-sided biased
dice are rolled (each die has 4 faces) and their absolute difference is used
to determine the number of tosses to make by a biased
coin (
Head
or
Tail
) in the next step (if the
difference is 0, then a single toss is made). In every trial, it is strictly necessary to stop the experiment,
if the result of the current toss is equal to that of the last toss, even if the required number of tosses is
not reached. Assuming that the ordered set of tosses is observed as an outcome (e.g. “…
HT
…”),
a)
Write the sample space for this experiment.
b)
Assuming that two subsets of the sample space are defined as
E
1
and
E
2
, write the smallest
field,
, which contains these two sets.
c)
Let
E
1
= (“
number of heads
number of tails
”) and
E
2
= (“
number of heads
1”). Find the
probability of set
A
= {
H
,
TH
,
HT
}, if the following event probabilities are given P(
E
1
) = 0.62
and P(
E
2
) = 0.73 (Hint
. Represent set
A
in terms of sets
E
1
and
E
2
.).
a)
,
,
,
,
,
,
,
,
,
T H TT HH TH HT THH HTT THT HTH
b)
1
1
2
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
,
,
,
,
,
,
,
,
,
,
,
,
,
E E
E E
E
E E
E
E
E E
E
E
E E
E
E
E E
E
c)
1
,
,
,
,
,
E
H HH TH HT THH HTH
2
,
,
,
,
,
,
E
T H TT TH HT HTT THT
12
A E
E
1
2
1
2
1
2
1
A
P E
E
P
P E
P E
P E
E
1
0.62
0.73 1
0.35
P A
P E
P E