What is the probability that a sum of 5 on the 2 dice will occur at least once?

Question

Use the formula for the probability of the complement of an event.

A pair of dice is rolled 3 times. What is the probability that a sum of 5 on the 2 dice will occur at least once? (Round your answer to three decimal places.)

answer is not 0.207, 0.978,0.361, or 11.111

Answer 1

chance is 11.111% if rolled once (1-8/9)=1/9= .111...=11.11%

roll the dice twice,odds increase to 1-(8/9)^2=1-64/81 = 17/81= 20.988%

roll 3 times,and it's higher probability = 1- (8/9)^3= about .297668= 29.767%

roll an infinite number of times and it will come up 5 at least once as 1-(8/9)^infinity= 1-0= 1

probability it comes up at least once= 1 minus probability it never comes up =1-(8/9)^n where n= number of times you roll the dice

Answer 2

Let us address the biggest issue here first:

a probability is between 0 and 1 (inclusive).

You cannot have a negative probability or a probability greater than 1.

If you ever get something like 11.111 as your "probability",

immediately conclude that something went wrong and try again.

When we roll two 6-sided dice, the number of possible outcomes is 6 x 6 = 36.

Out of those outcomes, the following pairs will give a sum of 5:

(1, 4); (2, 3); (3, 2); (4, 1).

So assuming that the dice are fair, that the outcomes are equally likely,

the probability of obtaining a sum of 5 when rolling 2 dice is:

4 / 36 = 1 / 9

The probability of NOT obtaining a sum of 5 when rolling 2 dice is:

1 - 1/9 = 9/9 - 1/9 = 8/9

When a pair of dice is rolled 3 times, the probability that a sum of 5 will NOT occur on any of the 3 rolls is:

8/9 * 8/9 * 8/9 = (8/9)³

Since the successive rolls are independent, to find the probability of a sequence of outcomes, we can multiply the probabilities of each outcome in the sequence.

In more details, "a sum of 5 will NOT occur on any of the 3 rolls" means "the 1st roll does NOT give a sum of 5 AND the 2nd roll does NOT give a sum of 5 AND the 3rd roll does NOT give a sum of 5" so we are looking at the intersection of these independent events, thus the multiplication of the probabilities.

"a sum of 5 on the 2 dice will occur at least once"

is the complement of

"a sum of 5 will NOT occur on any of the 3 rolls"

so its probability is given by:

1 - (8/9)³