34% because every time you roll the dice each side has a probability of 17%, so since there are numbers 5 and 6 you want to roll, there is a 34% you will get those numbers. the first roll doesn't have an affect on your next roll
Ok, if you roll the dice once - you'll get: 1, 2, 3, 4, 5 or 6. If you roll it twice, you can get one of these combinations: 1,1 - 1,2 - 1,3 - 1,4 - 1,5 - 1,6 2,1 - 2,2 - 2,3 - 2,4 - 2,5 - 2,6 3,1 - 3,2 - 3,3 - 3,4 - 3,5 - 3,6 4,1 - 4,2 - 4,3 - 4,4 - 4,5 - 4,6 5,1 - 5,2 - 5,3 - 5,4 - 5,5 - 5,6 6,1 - 6,2 - 6,3 - 6,4 - 6,5 - 6,6 Here there are 36 combinations in total. Your chance of getting a 5 and a 2 is 1/36. 5,2 only appears once in this list of possible combinations. 2,5 doesn't count (as 2 would come up first).
The binomial probability formula can be used to find the probability of a binomial experiment for a specific number of successes. It <em>does not</em> find the probability for a <em>range</em> of successes, as in this case.
The <em>range</em> "x≤4" means x = 0 <em>or</em> x = 1 <em>or </em>x = 2 <em>or</em> x = 3 <em>or</em> x = 4, so there are five different probability calculations to do.
To to find the total probability, we use the addition rule that states that the probabilities of different events can be added to find the probability for the entire set of events only if the events are <em>Mutually Exclusive</em>. The outcomes of a binomial experiment are mutually exclusive for any value of x between zero and n, as long as n and p don't change, so we're allowed to add the five calculated probabilities together to find the total probability.
The probability that x ≤ 4 can be written as P (X ≤ 4) or as P (X = 0 or X = 1 or X = 2 or X = 3 or X = 4) which means (because of the addition rule) that P(x ≤ 4) = P(x = 0) + P(x = 1) + P (x = 2) + P (x = 3) + P (x = 4)
Therefore, the probability of x<4 successes is P (X ≤ 4)
