Chapter 5, Problem R5.3RE

(a)

To determine

Probability model for this chance process.

Answer to Problem R5.3RE

Probability of each outcome is $\frac{1}{9}$ .

Explanation of Solution

Given information:

Each player can choose one of the three options:

Rock, paper, scissors

Let

(x, y) represent the outcomes of the game.

Where,

x: choice of Player 1

y: choice of Player 2

Then

Possible outcomes of the game are as follows:

(rock, rock), (rock, paper), (rock, scissors),

(paper, rock), (paper, paper), (paper, scissors),

(scissors, rock), (scissors, paper), (scissors, scissors)

Note that

There are 9 possible outcomes.

Whereas,

Each of the 9 outcomes is equal.

Thus,

The probability of each outcome is $\frac{1}{9}$ .

(b)

To determine

Probability for Player 1 wins the game on the first throw.

Answer to Problem R5.3RE

Probability that Player 1 wins the game on the first throw is approx. 0.3333.

Explanation of Solution

Given information:

Each player can choose one of the three options:

Rock, paper, scissors

Let

(x, y) represent the outcomes of the game.

Where,

x: choice of Player 1

y: choice of Player 2

Then

Possible outcomes of the game are as follows:

(rock, rock), (rock, paper), (rock, scissors),

(paper, rock), (paper, paper), (paper, scissors),

(scissors, rock), (scissors, paper), (scissors, scissors)

We know that

Rock wins from scissors,

Paper wins from rock,

And

Scissors wins from paper.

Then

The outcomes for Player 1 wins are as follows:

(rock, scissors),(paper, rock), (scissors, paper)

Note that

3 of the 9 outcomes result in Player 1 winning.

Thus,

In this case, the number of favorable outcomes is 3 and number of possible outcomes is 9.

If we divide number of favorable outcome by the number of possible outcome, we get the probability.

  $P\left(\text{Player}\text{1}\text{win}\right)=\frac{\text{Number}\text{of}\text{favorable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{3}{9}=\frac{1}{3}\approx 0.3333=33.33\%$