A professional gambler claims that he has loaded a die so that the outcomes of 1,2,3,4,5,6 have corrresponding probabilities of 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6. Can he actually do what he has claimed? Is a probablility distrubion described by listening the outcomes along with their corresponding probabilities?
The either-or probability is found by adding the individual probabilities. The probabilities of each outcome need to add to one.
What is the sum of your probabilities?
21/6
To determine if the professional gambler's claim is valid, we need to check if the probabilities provided align with the characteristics of a fair die.
A fair die is one where each outcome has an equal probability of occurring. In standard six-sided dice, each face has a probability of 1/6, which represents approximately 0.167 or 16.7% chance. If the probabilities of the outcomes on the die in question do not match these values, then it is not a fair die.
Here's how we can verify the gambler's claim by examining the probability distribution:
1. Write down the outcomes along with their corresponding probabilities provided by the gambler:
- Outcome 1: Probability 0.1
- Outcome 2: Probability 0.2
- Outcome 3: Probability 0.3
- Outcome 4: Probability 0.4
- Outcome 5: Probability 0.5
- Outcome 6: Probability 0.6
2. Calculate the sum of all the probabilities:
- Sum = 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 = 2.1
3. Check if the sum is equal to 1 (or very close to it considering rounding errors):
- In this case, the sum is 2.1, which is greater than 1.
Since the sum of the probabilities exceeds 1, we can conclude that the die described by the professional gambler is not valid or fair. This means they cannot actually load the die as claimed.
A probability distribution, as described here, consists of listing all possible outcomes along with their corresponding probabilities. It allows us to understand the likelihood of different events occurring and is commonly used in statistics and probability theory.