Probability of AT LEAST 3 6s in 5 Dice Rolls

• MHB
• Peter Mole
In summary: The probabilty of "exactly three sixes" is 0.03215+ 0.003215+ 0.0001286= 0.0354936.In summary, the best way to calculate the probability of at least three sixes occurring when rolling five six-sided dice is to add the probabilities of exactly three, four, and five sixes occurring. This can be done by calculating the probability of each occurrence separately and then adding the results.
Peter Mole
I've been teaching myself Probability Mathematics, but I'm still struggling.

Say I roll five six-sided dice all at once, with die faces numbered 1 through 6.
I want to determine the probability of AT LEAST three 6s occurring. First, is this the best way to set up the equation?

P(at least three 6s) = 1 - P(zero 6s) - P(exactly one 6) - P(exactly two 6s)

Or can the equation be better setup by subtracting AT LEAST occurrences?

I know this much...
P(zero 6s) = 5/6 * 5/6 * 5/6 * 5/6 * 5/6 = 3,125/7,776 = 0.40188
P(all 6s) = 1/6 * 1/6 * 1/6 *1/6 * 1/6 = 1/7,776
P(at least one 6) = 1 - P(zero 6s) = 4,651/7,776

To restate my whole problem, I'm unclear on the best way to calculate for "exactly 2" occurrences. Likewise for "exactly 3" or any other "exact" occurrence more than one but less than equal to the total number of dice thrown.

I wouldn't subtract. I would add. Rolling 5 dice, "at least three sixes" means 3 or 4 or 5. Calculate each of those separatelt then add them. On anyone toss, the probability of a "six" is 1/6, the probability of anything else is 5/6.

a) Exactly 3 sixes. The probability of 3 sixes and 2 non-sixes in that order is (1/6)(1/6)(1/6)(5/6)(5/6). Writing "s" for a six, "n" for a non-six, that would be "sssnn". But there are $\frac{5!}{3!2!}= 10$ different orders in which we can write "3 s's and 2 n's and each order has the same probability so there are 10(1/6)(1/6)(1/6)(5/6)(5/6)= [FONT=Verdana,Arial,Tahoma,Calibri,Geneva,sans-serif]0.0321502[/FONT][FONT=Verdana,Arial,Tahoma,Calibri,Geneva,sans-serif] approximately.

b) Exactly 4 sixes.
The probability of 4 sixes and 1 non-six in that order is (1/6)(1/6)(1/6)(1/6)(5/6). Writing "s" for a six, "n" for a non-six, that would be "ssssn". But there are $\frac{5!}{4!1!}= 5$ different orders in which we can write "3 s's and 2 n's and each order has the same probability so there are 5(1/6)(1/6)(1/6)(1/6)(5/6)= [FONT=Verdana,Arial,Tahoma,Calibri,Geneva,sans-serif]0.0032150 [/FONT]
[FONT=Verdana,Arial,Tahoma,Calibri,Geneva,sans-serif]
approximately.

c) Exactly 5 sixes. That is, of course, (1/6)(1/6)(1/6)(1/6)(1/6)= 0.0001286, approximately.

[/FONT]
[/FONT]

What is the probability of getting at least 3 6s in 5 dice rolls?

The probability of getting at least 3 6s in 5 dice rolls can be calculated using the binomial distribution formula. It is equal to 1 minus the probability of getting 0, 1, or 2 6s in 5 dice rolls.

How do you calculate the probability of getting at least 3 6s in 5 dice rolls?

To calculate the probability of getting at least 3 6s in 5 dice rolls, you can use the binomial distribution formula. This formula takes into account the number of trials (5), the probability of success (1/6 for getting a 6 on one roll), and the number of successes (3, 4, or 5).

What is the probability of getting exactly 3 6s in 5 dice rolls?

The probability of getting exactly 3 6s in 5 dice rolls can also be calculated using the binomial distribution formula. It is equal to the number of ways to get 3 6s (10) divided by the total number of possible outcomes (6^5).

How does the number of dice rolls affect the probability of getting at least 3 6s?

The more dice rolls you have, the higher the probability of getting at least 3 6s. This is because as the number of trials increases, the number of possible outcomes also increases, making it more likely to get at least 3 6s.

What other factors can affect the probability of getting at least 3 6s in 5 dice rolls?

The probability of getting at least 3 6s in 5 dice rolls can also be affected by the number of sides on the dice, the number of dice being rolled, and the method of rolling (e.g. rolling all 5 dice at once or rolling one at a time).

Replies
2
Views
3K
Replies
1
Views
1K
Replies
4
Views
1K
Replies
5
Views
1K
Replies
8
Views
4K
Replies
8
Views
2K
Replies
9
Views
2K
Replies
4
Views
2K
Replies
3
Views
2K
Replies
6
Views
4K