Question

A common way for two people to settle a frivolous dispute is to play a game of rock-paper-scissors. In this game, each person simultaneously displays a hand signal to Indicate a rock, a piece of paper, or a pair of scissors. Rock beats scissors, scissors beats paper, and paper beats rock. If both players select the same hand signal, the game results in a tie. Two roommates, roommate A and roommate B, are expecting company and are arguing over who should have to wash the dishes before the company arrives. Roommate A suggests a game of rock-paper-scissors to settle the dispute. Consider the game of rock-paper-scissors to be an experiment. In the long run, roommate A chooses rock 36% of the time, and roommate B chooses rock 22% of the time; roommate A selects paper 32% of the time, and roommate B selects paper 25% of the time; roommate A chooses scissors 32% of the time, and roommate B chooses scissors 53% of the time. (These choices are made randomly and independently of each other.) The probabilities were assigned using the_________.
Define event A as the event that roommate A wins the game and thus does not have to wash the dishes. What is P(A), the probability of event A?
P(A) = 0.32
P(A) = 0.22
P(A) = 0.34
P(A) = 0.38
Let event C be the event that the game ends in a tie. What is P(C), the probability of event C?
a. P(C) = 0.44
b. P(C) = 0.33
c. P(C) = 0.55
d. P(C) = 0.25
Define event B as the event that roommate B wins the game and thus does not have to wash the dishes.
What is the probability that roommate B wins the game?
a. P(B) = 0.67
b. P(B) = 0.43
c. P(B) = 0.33
d. P(B) = 0.50
What is the complement of event A?

Answer 1

Answer:

(a) P(A) = 0.34

(b) P(B) = 0.33

(c) P(C) = 0.33

(d) The Complement of event A = 1 - P(A) = 0.66

Step-by-step explanation:

We are given that in the long run, roommate A chooses rock 36% of the time, and roommate B chooses rock 22% of the time; roommate A selects paper 32% of the time, and roommate B selects paper 25% of the time; roommate A chooses scissors 32% of the time, and roommate B chooses scissors 53% of the time.

Let the probability that roommate A chooses rock = P($R_A$) = 0.36

The probability that roommate A chooses paper = P($P_A$) = 0.32

The probability that roommate A chooses scissors = P($S_A$) = 0.32

The probability that roommate B chooses rock = P($R_B$) = 0.22

The probability that roommate B chooses paper = P($P_B$) = 0.25

The probability that roommate B chooses scissors = P($S_B$) = 0.53

(a) Let A = event that roommate A wins the game and thus does not have to wash the dishes.

This will happen only when roommate A chooses rock and roommate B chooses scissors or roommate A chooses paper and roommate B chooses rock or roommate A chooses scissors and roommate B chooses paper.

So, P(A) =  $P(R_A) \times P(S_B) + P(P_A) \times P(R_B) + P(S_A) \times P(P_B)$  

             =  (0.36 $\times$ 0.53) + (0.32 $\times$ 0.22) + (0.32 $\times$ 0.25)  

             =  0.1908 + 0.0704 + 0.08

    P(A)  =  0.34

(b) Let B = event that roommate B wins the game and thus does not have to wash the dishes.

This will happen only when roommate B chooses rock and roommate A chooses scissors or roommate B chooses paper and roommate A chooses rock or roommate B chooses scissors and roommate A chooses paper.

So, P(B) =  $P(R_B) \times P(S_A) + P(P_B) \times P(R_A) + P(S_B) \times P(P_A)$  

             =  (0.22 $\times$ 0.32) + (0.25 $\times$ 0.36) + (0.53 $\times$ 0.32)  

             =  0.0704 + 0.09 + 0.1696

    P(B)  =  0.33

(c) Let C = event that the game ends in a tie.

This will happen only when roommate A chooses rock and roommate B also chooses rock or roommate A chooses paper and roommate B also chooses paper or roommate A chooses scissors and roommate B also chooses scissors.

So, P(C) =  $P(R_A) \times P(R_B) + P(P_A) \times P(P_B) + P(S_A) \times P(S_B)$  

             =  (0.36 $\times$ 0.22) + (0.32 $\times$ 0.25) + (0.32 $\times$ 0.53)  

             =  0.0792 + 0.08 + 0.1696

    P(C)  =  0.3288 ≈ 0.33

(d) The complement of event A = P(A') = 1 - P(A)

                                                       = 1 - 0.34 = 0.66.