Ulysses and Iolanthe are going to play checkers. They enjoy playing “matches” wh

Question

Ulysses and Iolanthe are going to play checkers. They enjoy playing “matches” wherein the first to win two games wins the match. In each game either Ulysses wins or Iolanthe wins; that is, there are no “draws”. It turns out that they are evenly matched players, but there is an advantage to moving first. So the player who goes first wins 52% of the games. They flip a coin to determine who goes first in the first game, and Ulysses wins. He will move first in the first and third games (if the third game is needed for the match), while Iolanthe will go first in the second game. For this problem we assume that the games are independent
a. Using the notation UU to indicate that Ulysses won both the first and second games; and UII to indicate that Ulysses won the first game, but Iolanthe won the second and third games, write the sample space for the match.
b. Calculate the probabilities for all the events in the sample space, and give a table indicating the probability mass function for the match.
c. Let X represent the number of games Ulysses wins in the match. Using your answer to part b, give the probability mass function for X.
d. Let Y represent the number of games Iolanthe wins in the match. Using your answer to part b, give the probability mass function for Y .
e. Let Z represent the number of games played in the match. Using your answer to part b, give the probability mass function for Z.
f. Let A represent the winner of the match. Using your answers to b, c, d, or e, as appropriate, give the probability mass function for A.
g. Had Iolanthe won the coin toss and gone first in the first and third games, what is the probability she would have won the match? Ulysses and Iolanthe are going to play checkers. They enjoy playing “matches” wherein the first to win two games wins the match. In each game either Ulysses wins or Iolanthe wins; that is, there are no “draws”. It turns out that they are evenly matched players, but there is an advantage to moving first. So the player who goes first wins 52% of the games. They flip a coin to determine who goes first in the first game, and Ulysses wins. He will move first in the first and third games (if the third game is needed for the match), while Iolanthe will go first in the second game. For this problem we assume that the games are independent
a. Using the notation UU to indicate that Ulysses won both the first and second games; and UII to indicate that Ulysses won the first game, but Iolanthe won the second and third games, write the sample space for the match.
b. Calculate the probabilities for all the events in the sample space, and give a table indicating the probability mass function for the match.
c. Let X represent the number of games Ulysses wins in the match. Using your answer to part b, give the probability mass function for X.
d. Let Y represent the number of games Iolanthe wins in the match. Using your answer to part b, give the probability mass function for Y .
e. Let Z represent the number of games played in the match. Using your answer to part b, give the probability mass function for Z.
f. Let A represent the winner of the match. Using your answers to b, c, d, or e, as appropriate, give the probability mass function for A.
g. Had Iolanthe won the coin toss and gone first in the first and third games, what is the probability she would have won the match? Ulysses and Iolanthe are going to play checkers. They enjoy playing “matches” wherein the first to win two games wins the match. In each game either Ulysses wins or Iolanthe wins; that is, there are no “draws”. It turns out that they are evenly matched players, but there is an advantage to moving first. So the player who goes first wins 52% of the games. They flip a coin to determine who goes first in the first game, and Ulysses wins. He will move first in the first and third games (if the third game is needed for the match), while Iolanthe will go first in the second game. For this problem we assume that the games are independent
a. Using the notation UU to indicate that Ulysses won both the first and second games; and UII to indicate that Ulysses won the first game, but Iolanthe won the second and third games, write the sample space for the match.
b. Calculate the probabilities for all the events in the sample space, and give a table indicating the probability mass function for the match.
c. Let X represent the number of games Ulysses wins in the match. Using your answer to part b, give the probability mass function for X.
d. Let Y represent the number of games Iolanthe wins in the match. Using your answer to part b, give the probability mass function for Y .
e. Let Z represent the number of games played in the match. Using your answer to part b, give the probability mass function for Z.
f. Let A represent the winner of the match. Using your answers to b, c, d, or e, as appropriate, give the probability mass function for A.
g. Had Iolanthe won the coin toss and gone first in the first and third games, what is the probability she would have won the match?

Explanation / Answer

(a) Sample Space : {UU, LL, ULL, ULU, LUU,LUL}

(b) Pr(UU) = 0.52 * 0.48 = 0.2496

Pr(LL) = 0.48 * 0.52 = 0.2496

Pr(ULL) = 0.52 * 0.52 * 0.48 = 0.1298

Pr(ULU) = 0.52 * 0.52 * 0.52 = 0.1406

Pr(LUU) = 0.48 * 0.48 * 0.52 = 0.1198

Pr(LUL) = 0.48 * 0.48 * 0.48 = 0.1106

(c) X can have value from 0,1 and 2

so If X = 0 for LL

X = 1 for ULL, LUL

X = 2 for UU, ULU and LUU

so P(X) = 0.2496 ; X = 0

= 0.1298 + 0.1106 = 0.2404 ; X =1

= 0.2496 + 0.1406 + 0.1198 = 0.51 ; X = 2

(d) Y can have values from 0,1 and 2

Y = 0 for UU

Y = 1 for LUU, ULU

Y = 2 for LL,ULL, LUL

P(Y) = 0.2496 ; Y = 0

=0.1406 + 0.1198 = 0.2604 ; Y =1

= 0.2496 + 0.1298 + 0.1106 = 0.49 ; Y = 2

(e) Z = 2 or 3 as there are either 2 games player or 3 games played

so Z = 2 when UU and LL

Z = 3 ; otherwise

f(Z) = 2 * 0.2496 = 0.4992 ; Z = 2

= 0.1298 + 0.1106 + 0.1406 + 0.1198  = 0.5008 ; Z = 3

(f) A can have sample space is U win and L win

Pr(U win) at UU, ULU and LUU

Pr(U win) = 0.2496 + 0.1406 + 0.1198 = 0.51

Pr(L win) = 0.2496 +  0.1298 + 0.1106 = 0.41

(g) Now loanthe would take the place of Ulyssis so

Pr(Loanthe would have win) = 0.51

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