What is the difference between these two probability problems?

In summary, the conversation discusses two probability questions involving rolling dice and getting a sum of 20. The difference between the two problems is the method used to solve them, as well as the order of the dice rolls not affecting the outcome in the first problem. To solve these problems, one can use modern technology or paper and pencil methods to count the number of events with a sum of 20.
  • #1
davedave
50
0
I happened to come across these two probability questions in a library book.

1) Suppose that you roll 5 six-sided dice at the same time. What is the probability of getting
a sum of 20?

2) Suppose that you roll a six-sided die 5 times. What is the probability of getting a sum of
20 on the 5 rolls?

What is the difference between these two problems?

What methods are used to solve them?
 
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  • #2
davedave said:
What is the difference between these two problems?

Good question. If the dice are identical then the final answer will be the same because the order of the dice rolls doesn't affect the outcome of the sum. The event space of dice rolls can be written as {1,2,...,6}^5 = {(1,1,1,1,1),(1,1,1,1,2),...,(6,6,6,6,6)} with all events equally probable, so you'll need to find a way of counting the number of events with sum 20.
 
  • #3
bpet said:
Good question. If the dice are identical then the final answer will be the same because the order of the dice rolls doesn't affect the outcome of the sum. The event space of dice rolls can be written as {1,2,...,6}^5 = {(1,1,1,1,1),(1,1,1,1,2),...,(6,6,6,6,6)} with all events equally probable, so you'll need to find a way of counting the number of events with sum 20.
An easy way (given modern technology) of counting the number of events that sum to 20 is to observe that it is the coefficient of x^20 in (x + x^2 + x^3 + x^4 + x^5 + x^6)^5.

Just type "expand (x + x^2 + x^3 + x^4 + x^5 + x^6)^5" into Wolfram Alpha and hit Enter.

You can also find the coefficient of x^20 by paper and pencil methods-- it's not that hard, just a little more work.
 
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FAQ: What is the difference between these two probability problems?

What is the difference between these two probability problems?

1. What is probability?

Probability is a measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 represents impossibility and 1 represents certainty.

2. How is probability different from odds?

Probability and odds are related concepts, but they are not the same. Probability is the likelihood of an event occurring, while odds are the ratio of the probability of an event occurring to the probability of it not occurring. For example, if the probability of an event occurring is 0.5, the odds would be 1:1 or "even odds".

3. What is the difference between dependent and independent events?

In probability, events are considered dependent if the outcome of one event affects the outcome of another event. Independent events, on the other hand, are events where the outcome of one event has no effect on the outcome of another event.

4. What is the difference between conditional and unconditional probability?

Conditional probability is the probability of an event occurring given that another event has already occurred. Unconditional probability, also known as marginal probability, is the probability of an event occurring without any conditions or restrictions.

5. How do you calculate probability?

Probability is typically calculated by dividing the number of favorable outcomes by the total number of possible outcomes. This is known as the classical probability approach. Other approaches to calculating probability include empirical probability, which is based on observed data, and subjective probability, which is based on personal beliefs or opinions.

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