Question

cConsider a game in which players roll a number cube to determine the number of points earned. If a player rolls a prime number, that many points will be added to the player’s total. Any other roll will be deducted from the player’s total. What is the expected value of the points earned on a single roll in this game?

Answer 1

E(X) = 1/6

Explanation:

Given:-

- A numbered cube is rolled.

+ Points : Prime Number

- Points : Any other number.

Find:-

What is the expected value of the points earned on a single roll in this game?

Solution:-

- We will denote a random variable (X) that represents the number of points in a game. We have 6 faces with (1 , 2 , 3 , 5) as prime numbers ( + Points ). Where numbers ( 4 & 6 ) will denote the ( - Points ).

- We will express the random variable (X) in probability distribution table. The probability of getting any number on the cube is equal to (1/6). The distribution table is as follows:

X -6 -4 1 2 3 5

P(X) 1/6 1/6 1/6 1/6 1/6 1/6

- The expected value for the random variable E(X) can be determined by the following formula:

E(X) = Sum ( Xi*P(X) )

Where, i : Term number from table:

E(X) = (-6)*(1/6) + (-4)*(1/6) + (1)*(1/6) + (2)*(1/6) + (3)*(1/6) + (5)*(1/6)

E(X) = (1/6)* ( -6 -4 + 1 + 2 + 3 + 5 )

E(X) = ( 1 / 6 ) * ( 1 )

E(X) = 1/6

- The expected value of points earned per single roll in this game is 1/6

Answer 2

Expected value = -1(1/6) + 2(1/6) + 3(1/6) - 4(1/6) + 5(1/6) - 6(1/6) = -1/6 + 2/6 + 3/6 - 4/6 + 5/6 - 6/6 = -1/6 = -0.167