Probability question considering 6 dice throws

In summary, the conversation discusses the probability of rolling numbers 5 and 6 at least once in six throws of a dice. The calculation involves finding the probability of not rolling a 5 or 6 in a single throw and then using that to calculate the probability for all six throws. The final answer is approximately 41.8%.
  • #1
Uniquebum
55
1

Homework Statement


Dice is thrown 6 times. What's the probability of numbers 5 and 6 showing up at least once.

Homework Equations


This ought to be basic probability calculus but i just can't get my head around this. Some kind of attempt(ish) below. THe answer ought to be 0.418 or 41.8%.

The Attempt at a Solution


Now, i know the amount of different permutations is 6^6. Suppose A = [a number 5 appears] and B = [a number 6 appears].
I think i'd need to do this as
P(A n B) = 1 - P(A^c) - P(B^c) + P(A^c n B^c) where ^c denotes a complement.

Probability of A being false (5 not showing) = (5/6)^6 and it's the same for B. I just can't figure out what P(A^c n B^c) is... It's "something" divided by 6^6 but that's as far as i get... Any help would be appreciated.
 
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  • #2
A and B false <=> no 5 and no 6.
Can you calculate the probability of "not 5 and not 6" for a single throw? With that probability, it is easy to calculate "no 5 and no 6" for all 6 throws.
 
  • #3
Thanks alot! Got it done :). Final calculation being

P(A n B) = 1 - (5/6)^6 - (5/6)^6 + (4/6)^6 = 0.41799...

Anyway, thanks again!
 
  • #4
Wouldn't the probability be 1 - (4/6)^6 = .9122?

Since the probability of rolling any other number is (4/6) for one roll. And for six rolls it is (4/6)^6.
 
  • #5
Biosyn said:
Wouldn't the probability be 1 - (4/6)^6 = .9122?

Since the probability of rolling any other number is (4/6) for one roll. And for six rolls it is (4/6)^6.
No, the question aks for the probability of rolling at least one 5 and at least one 6, not at least one 5 or 6.
 

1. What is the probability of rolling a specific number with 6 dice throws?

The probability of rolling a specific number (such as a 6) with 6 dice throws is 1/6, or about 16.67%. This is because each dice throw has an equal chance of landing on any of the 6 numbers.

2. What is the probability of rolling a specific combination of numbers with 6 dice throws?

The probability of rolling a specific combination of numbers (such as 3 sixes and 3 ones) with 6 dice throws depends on the combination itself. For example, the probability of rolling 3 sixes and 3 ones is (1/6)^3 * (1/6)^3 = 1/46,656, or about 0.002%.

3. What is the probability of rolling a total sum of 20 with 6 dice throws?

The probability of rolling a total sum of 20 with 6 dice throws depends on the specific combination of numbers that add up to 20. This probability can be calculated by considering all possible combinations and dividing by the total number of outcomes (6^6 = 46,656).

4. Can the probability of rolling a specific number increase with more dice throws?

Yes, the probability of rolling a specific number can increase with more dice throws. For example, the probability of rolling a specific number with 6 dice throws is 1/6, but with 12 dice throws, it increases to 1/12. This is because there are more chances for the desired number to appear with more dice throws.

5. How does the probability of rolling a specific number change if one or more dice are removed?

The probability of rolling a specific number changes if one or more dice are removed because there are now fewer possible outcomes. For example, if one dice is removed from the previous scenario of 12 dice throws, the probability of rolling a specific number now becomes 1/11 instead of 1/12. This is because there are now 11 possible outcomes instead of 12.

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