Answer:
(a) The PMF of <em>X</em> is:
(b) The probability that a player defeats at least two opponents in a game is 0.64.
(c) The expected number of opponents contested in a game is 5.
(d) The probability that a player contests four or more opponents in a game is 0.512.
(e) The expected number of game plays until a player contests four or more opponents is 2.
Step-by-step explanation:
Let <em>X</em> = number of games played.
It is provided that the player continues to contest opponents until defeated.
(a)
The random variable <em>X</em> follows a Geometric distribution.
The probability mass function of <em>X</em> is:
It is provided that the player has a probability of 0.80 to defeat each opponent. This implies that there is 0.20 probability that the player will be defeated by each opponent.
Then the PMF of <em>X</em> is:
(b)
Compute the probability that a player defeats at least two opponents in a game as follows:
P (X ≥ 2) = 1 - P (X ≤ 2)
= 1 - P (X = 1) - P (X = 2)
Thus, the probability that a player defeats at least two opponents in a game is 0.64.
(c)
The expected value of a Geometric distribution is given by,
Compute the expected number of opponents contested in a game as follows:
Thus, the expected number of opponents contested in a game is 5.
(d)
Compute the probability that a player contests four or more opponents in a game as follows:
P (X ≥ 4) = 1 - P (X ≤ 3)
= 1 - P (X = 1) - P (X = 2) - P (X = 3)
Thus, the probability that a player contests four or more opponents in a game is 0.512.
(e)
Compute the expected number of game plays until a player contests four or more opponents as follows:
Thus, the expected number of game plays until a player contests four or more opponents is 2.