Tag Archives: JCM_math230_HW1_S15

Consider the probability space

\[ \Omega = \{ HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\}\]

as the outcome of three consecutive tosses of a coin. (We make the reasonable assumption that all outcomes are are equally likely.) The event

\[ \{ HHH,TTT\}\]

is the event that all three tosses have the same outcome. Give a similar verbal description to each of the events bellow:

1. \(\{HHH,HHT,HTH,HTT\}\)
2. \(\{HTH,HTT,TTT,TTH\}\)
3. \(\{HTT,HTH,HHT,HHH\}\)
4. \(\{HTH,THH,TTH\}\)
5. \(\{THT,HTT,TTH\}\)
6. \(\{TTT,TTH,THT,HTT\}\)
7. \(\{HHT,HHH,TTH,TTT\}\)

[Pitman, p31, #5] (Assign 1 and 5 first).

Events \(A\), \(B\), and \(C\) are defined on a probability space. Find expressions for the
following probabilities in terms of \(\mathbf{P}(A)\), \(\mathbf{P}(B)\), \(\mathbf{P}(C)\), \(\mathbf{P}(AB)\), \(\mathbf{P}(AC)\), \(\mathbf{P}(BC)\), and \(\mathbf{P}(ABC)\).

1. The probability that exactly two of the \(A\), \(B\), \(C\) occur.
2. The probability that exactly one of these events occurs.
3. The probability that none of these events occur.

Here the notation \(AB\) is short for \(A \cap B\) which is the event “both \(A\) and \(B\)”

( [Pitman, p. 31, # 10])