Probability dice roll questions

In summary, the conversation is discussing a problem where a player rolls a fair die and receives a number of dollars equal to the number of dots on the face of the die. The question is what is the least amount the player should expect to pay in order to play the game. The solution involves finding the expected value of the random variable and determining if it is a relevant "fair price" for a one-shot game. There is also a discussion about the difference between expected value and average, which can be ambiguous.
  • #1
mtingt
13
0

Homework Statement



A player rolls a fair die and receives a number of dollars equal to the number of dots appearing on the face of the die. What is the least a player should expect to pay in order to play the game?


Homework Equations



I don't really understand the question?? can someone explain the questions or approach thanks

The Attempt at a Solution

 
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  • #2
I think it means what would the host of the game need to receive from each player in order to not win or lose any money in the long run. Players are going to be paid anywhere between 1 to 6 dollars each game they play, so I'm guessing the answer is the average?

I could be way off though.
 
  • #3
I've actually seen this problem once or twice and in the end it is the average.

To see this, you just need to write down what the problem says:

$(1/6) + $(1/6) + $(1/6) + $ (1/6) + $(1/6) + $(1/6)

The $ represent the amount of money the person would get if a particular number is shown so for example, if 1 dot is shown the player gets 1 dollar, if 5 dots are shown the player gets 5 dollars, and so on and so forth.
 
  • #4
In a probability class they would say that the roll of a die is a random variable. You are asked to find the expected value of the random variable.

The expected value does turn out to be the average in this case, but that is not true for all random variables.
 
  • #5
awkward said:
In a probability class they would say that the roll of a die is a random variable. You are asked to find the expected value of the random variable.

The expected value does turn out to be the average in this case, but that is not true for all random variables.

This statement is highly misleading and likely wrong. For a single random variable, the average and the expected value are essentially the same, BY DEFINITION. If you have some other type of situation in mind, please tell us what it is. (Of course, in more advanced courses one can encounter random variables that do not have expected values, etc., but this would only be confusing to mention at all to a person taking a first course at an elementary level.)

A much mores serious objection is related to the issue of whether or not the expected value is really a relevant "fair price" in a one-shot situation (that is, in a situation in which you play the game just once). From the point of view of a casino, the expected value makes sense because the casino faces thousands of players and so is, in effect, playing many times. However, for an individual customer, the expected value may be a very poor way to decide whether the entry fee is too high or just right. All of this is connected to expected utility ideas.

RGV
 
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  • #6
Ray Vickson,

I take it you are disagreeing with my statement that the expected value is not, in general, the average? If so, allow me to point out that the previous posts in the thread seemed to be using "average" in the sense of the arithmetic mean of the values of the random variable. That is the sense I meant to refer to. But if you define "average" to mean "expected value", of course, that is a different proposition.

Sorry for any confusion, I should have been more explicit. "Average" is one of those ambiguous words.
 
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  • #7
awkward said:
Ray Vickson,

I take it you are disagreeing with my statement that the expected value is not, in general, the average? If so, allow me to point out that the previous posts in the thread seemed to be using "average" in the sense of the arithmetic mean of the values of the random variable. That is the sense I meant to refer to. But if you define "average" to mean "expected value", of course, that is a different proposition.

Sorry for any confusion, I should have been more explicit. "Average" is one of those ambiguous words.

I agree, and that is why I was careful to say "for a single random variable..."; however, the way I read you post, you were also talking about a single random variable, but maybe you really meant something different.

RGV
 

FAQ: Probability dice roll questions

1. What is the probability of rolling a specific number on a single six-sided dice?

The probability of rolling a specific number on a single six-sided dice is 1/6 or approximately 16.67%. This is because there are six possible outcomes (numbers 1-6) and only one of those outcomes will result in the specific number being rolled.

2. What is the probability of rolling a certain sum with two six-sided dice?

The probability of rolling a certain sum with two six-sided dice depends on the sum you are trying to achieve. For example, the probability of rolling a sum of 7 is 1/6 or approximately 16.67%, as there are six possible combinations that can result in a sum of 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). However, the probability of rolling a sum of 2 or 12 is much lower at 1/36 or approximately 2.78%, as there is only one possible combination for each of those sums (1+1 and 6+6, respectively).

3. What is the probability of rolling a specific number on multiple dice rolls?

The probability of rolling a specific number on multiple dice rolls is dependent on the number of dice rolls and the number of dice being rolled. For example, if you roll two six-sided dice, the probability of rolling a specific number on the first roll is 1/6. However, the probability of rolling that same number on the second roll is still 1/6, but the overall probability of rolling that specific number on either the first or second roll is 1/3. This is because there are two possible outcomes for each roll (the specific number or any other number) and the probability of either outcome occurring is 1/6.

4. How does the number of sides on a dice affect the probability of rolling a specific number?

The number of sides on a dice directly affects the probability of rolling a specific number. The more sides a dice has, the lower the probability of rolling a specific number. For example, a ten-sided dice will have a probability of 1/10 or approximately 10% for rolling a specific number, while a twenty-sided dice will have a probability of 1/20 or approximately 5%. This is because the more sides a dice has, the more possible outcomes there are, making it less likely for a specific number to be rolled.

5. Can the probability of rolling a specific number ever be higher than 1?

No, the probability of rolling a specific number cannot be higher than 1. This is because the probability of an event occurring is always a fraction or decimal between 0 and 1, with 0 representing impossibility and 1 representing certainty. The probability of rolling a specific number is always a fraction or decimal, making it impossible for it to be higher than 1.

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