# Expected Value Word Problems

### Solutions

1A pair of die is thrown. The random variable, X, is defined as the sum of the obtained scores. Determine the probability distribution, the expected value and variance.

2A player throws a die. If a prime number is obtained, he gains to win an amount equal to the number rolled times 100 dollars, but if a prime number is not obtained, he loses an amount equal to the number rolled times 100 dollars. Calculate the probability distribution and the expected value of the described game.

3The first prize for a raffle is $5,000 (with a probability of 0.001) and the second prize is $2,000 (with a probability of 0.003). What is a fair price to pay for a single ticket in this raffle?

4Let X be a discrete random variable whose probability distribution is as follows:

x | p _{i} |
---|---|

0 | 0,1 |

1 | 0,2 |

2 | 0,1 |

3 | 0,4 |

4 | 0,1 |

5 | 0,1 |

1. Calculate the distribution function.

2. Calculate the following probabilities:

p (X < 4.5)

p (X ≥ 3)

p (3 ≤ X < 4.5)

5A player tosses two coins into the air. He gains to win $1 times the number of heads that are obtained. However, he will lose $5 if neither coin is a head. Calculate the expected value of this game and determine whether it is favorable for the player.

6Knowing that p(X ≤ 2) = 0.7 and p(X ≥ 2) = 0.75. Calculate:

1. The expected value.

2.The variance.

3.The standard deviation.

## 1

A pair of die is thrown. The random variable, X, is defined as the sum of the obtained scores. Determine the probability distribution, the expected value and variance.

x | p _{i} |
x · p _{i} |
x^{ 2}· p_{i} |
---|---|---|---|

2 | 1/36 | 2/36 | 4/36 |

3 | 2/36 | 6/36 | 18/36 |

4 | 3/36 | 12/36 | 48/36 |

5 | 4 /36 | 20/3 6 | 100/36 |

6 | 5/36 | 30/36 | 180/36 |

7 | 6/36 | 42/36 | 294/36 |

8 | 5/36 | 40/36 | 320/36 |

9 | 4 /36 | 36/36 | 324/36 |

10 | 3/36 | 30/36 | 300/36 |

11 | 2/36 | 22/36 | 242/36 |

12 | 1/36 | 12/36 | 144/36 |

7 |
54.83 |

## 2

A player throws a die. If a prime number is obtained, he gains to win an amount equal to the number rolled times 100 dollars, but if a prime number is not obtained, he loses an amount equal to the number rolled times 100 dollars. Calculate the probability distribution and the expected value of the described game.

x | p _{i} |
x· p _{i} |
---|---|---|

+100 | 100/6 | |

+ 200 | 200/6 | |

+ 300 | 300/6 | |

- 400 | -400/6 | |

+ 500 | 500/6 | |

-600 | - 600/6 | |

100/6 |

µ =16.667

## 3

The first prize for a raffle is $5,000 (with a probability of 0.001) and the second prize is $2,000 (with a probability of 0.003). What is a fair price to pay for a single ticket in this raffle?

μ = 5000 · 0.001 + 2000 · 0.003 = $11

## 4

Let X be a discrete random variable whose probability distribution is as follows:

x | p _{i} |
---|---|

0 | 0,1 |

1 | 0,2 |

2 | 0,1 |

3 | 0,4 |

4 | 0,1 |

5 | 0,1 |

1. Calculate the distribution function.

2. Calculate the following probabilities:

p (X < 4.5)

p (X < 4.5) = F (4.5) = 0.9

p (X ≥ 3)

p (X ≥ 3) = 1 - p(X < 3) = 1 - 0.4 = 0.6

p (3 ≤ X < 4.5)

p (3 ≤ X < 4.5) = p (X < 4.5) - p(X < 3) = 0.9 - 0.4 = 0.5

## 5

A player tosses two coins into the air. He gains to win $1 times the number of heads that are obtained. However, he will lose $5 if neither coin is a head. Calculate the expected value of this game and determine whether it is favorable for the player.

E = {(c,c);(c,x);(x,c);(x,x)}

p(+1) = 2/4

p(+2) = 1/4

p(−5) = 1/4

μ = 1 · 2/4 + 2 · 1/4 - 5 · 1/4 = −1/4. It is unfavorable

## 6

Knowing that p(X ≤ 2) = 0.7 and p(X ≥ 2) = 0.75. Calculate:

The expected value, the variance and the standard deviation.

x | p _{i} |
x · p _{i} |
x^{ 2}· p_{i} |
---|---|---|---|

0 | 0.1 | 0 | 0 |

1 | 0.15 | 0.15 | 0.15 |

2 | 0.45 | 0.9 | 1.8 |

3 | 0.1 | 0.3 | 0.9 |

4 | 0.2 | 0.8 | 3.2 |

2.15 |
6.05 |

μ =2.15

σ² = 6.05 - 2.15² = 1.4275

σ = 1.19