Rolling 5 Dice: Probability of At Least 3 Sixes

In summary: This means that the probability of getting at least three sixes is 23/648. In summary, the probability of getting at least three sixes when rolling five standard 6-sided dice is 23/648. This can be found by using the complements rule and subtracting the probability of getting 0, 1, or 2 sixes from 1.
  • #1
veronica1999
61
0
Five standard 6-sided dice are rolled. What is the probability that at least 3 of them show a six?

I am surprised my answer is wrong.

First the total outcomes are 6x6x6x6x6=7776

Successful outcomes 10x6x6=360
There are 10 ways to choose 3 from 5 and then the remaining 2 can be any number.

360/7776 is not the answer.

The answer is 276/7776.:confused:
 
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  • #2
Re: dice problem

veronica1999 said:
Five standard 6-sided dice are rolled. What is the probability that at least 3 of them show a six?

I am surprised my answer is wrong.

First the total outcomes are 6x6x6x6x6=7776

Successful outcomes 10x6x6=360
There are 10 ways to choose 3 from 5 and then the remaining 2 can be any number.

360/7776 is not the answer.

The answer is 276/7776.:confused:
You have double-counted the ways in which four or five sixes can occur. What you should do is to count the number of ways in which exactly three sixes can occur (10x5x5), then add the number of ways in which four sixes can occur (5x5), and finally the one way in which five sixes can occur.
 
  • #3
Re: dice problem

Oops...
Thanks!:D
 
  • #4
Re: dice problem

I would use the complements rule:

\(\displaystyle P(X)=1-\frac{{5 \choose 0}5^5+{5 \choose 1}5^4+{5 \choose 2}5^3}{6^5}=\frac{23}{648}\)
 
  • #5


Hello,

Thank you for sharing your solution. I see that you have correctly calculated the total number of outcomes and the number of successful outcomes. However, the probability of at least 3 sixes is not just the number of successful outcomes divided by the total number of outcomes.

To calculate the probability of at least 3 sixes, we need to consider all possible combinations of 3, 4, or 5 sixes. This includes 3 sixes and 2 non-six numbers, 4 sixes and 1 non-six number, and 5 sixes and 0 non-six numbers. Therefore, the total number of successful outcomes is 360 + 120 + 5 = 485.

The probability of at least 3 sixes can be calculated as 485/7776, which is equivalent to 276/7776 after simplification. So, your final answer is correct. It is important to consider all possible combinations when calculating probabilities, rather than just the number of successful outcomes.

I hope this helps clarify any confusion. Keep up the good work in your calculations!
 

1. What is the probability of rolling at least 3 sixes with 5 dice?

The probability of rolling at least 3 sixes with 5 dice is approximately 0.3086, or about 31%. This means that if you were to roll 5 dice many times, you can expect to get at least 3 sixes about 31% of the time.

2. How do you calculate the probability of at least 3 sixes with 5 dice?

To calculate the probability of at least 3 sixes with 5 dice, you can use the binomial probability formula: P(x ≥ k) = 1 - P(x < k). In this case, x represents the number of sixes rolled and k is the minimum number of sixes needed (3 in this case). You will also need to use the combination formula to determine the number of ways to get 3 or more sixes out of 5 dice.

3. What is the expected number of sixes when rolling 5 dice?

The expected number of sixes when rolling 5 dice is 1.25. This means that on average, you can expect to get about 1 or 2 sixes when rolling 5 dice. However, the actual number of sixes you get may vary each time you roll the dice.

4. Is it possible to get more than 3 sixes when rolling 5 dice?

Yes, it is possible to get more than 3 sixes when rolling 5 dice. However, the probability of getting more than 3 sixes is much lower than the probability of getting at least 3 sixes. For example, the probability of getting exactly 4 sixes is about 0.0772, or about 8%.

5. How does the probability of rolling at least 3 sixes change if you add more dice?

The probability of rolling at least 3 sixes increases as you add more dice. For example, if you were to roll 6 dice instead of 5, the probability of getting at least 3 sixes would increase to approximately 0.4213, or about 42%. This is because the more dice you have, the more chances you have of getting a six.

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