A prize of $100, $200, $500, or $1000 will be randomly awarded to a game show contestant, with the probabilities of winning each prize shown in the table below. What is the expected value of the prize that will be awarded?
no table. But multiply each amount by its probability, and add up the results.
To find the expected value of the prize that will be awarded, we need to calculate the weighted average of the prizes, where each prize is multiplied by its probability of being awarded.
Let's denote the prizes as follows:
$100 as X₁,
$200 as X₂,
$500 as X₃,
$1000 as X₄.
The probabilities of winning each prize are given in the table below:
Prize (X) | Probability (P)
------------------------------
X₁ | 0.4
X₂ | 0.3
X₃ | 0.2
X₄ | 0.1
To find the expected value, we multiply each prize by its corresponding probability, and then sum them up:
Expected Value = (X₁ * P₁) + (X₂ * P₂) + (X₃ * P₃) + (X₄ * P₄)
Expected Value = ($100 * 0.4) + ($200 * 0.3) + ($500 * 0.2) + ($1000 * 0.1)
Expected Value = $40 + $60 + $100 + $100
Expected Value = $300
Therefore, the expected value of the prize that will be awarded is $300.
To determine the expected value of the prize that will be awarded, we multiply each prize by its respective probability and then sum the results.
Let's calculate the expected value step by step:
1. Calculate the product of each prize and its probability:
- Product of $100 and probability 0.4: $100 * 0.4 = $40
- Product of $200 and probability 0.3: $200 * 0.3 = $60
- Product of $500 and probability 0.2: $500 * 0.2 = $100
- Product of $1000 and probability 0.1: $1000 * 0.1 = $100
2. Sum up the results:
$40 + $60 + $100 + $100 = $300
The expected value of the prize that will be awarded is $300.