Final answer:
The probability of rolling a sum of 5 at least four times in five throws of a pair of fair dice is calculated using the binomial probability formula and is approximately 2/19683.
Step-by-step explanation:
The question concerns the probability of throwing a sum of 5 at least 4 times in 5 throws of a pair of fair dice. To solve this, we can use the binomial probability formula, which is P(X=k) = (n choose k) * p^k * (1-p)^(n-k), where 'n' is the number of trials, 'k' is the number of successes, 'p' is the probability of success on a single trial, and '1-p' is the probability of failure on a single trial.
There are four outcomes when throwing a pair of fair dice that sum to 5: (1,4), (2,3), (3,2), and (4,1). Since there are 6 * 6 = 36 possible outcomes when rolling two dice, the probability of rolling a sum of 5 on a single throw is 4/36, which simplifies to 1/9.
To calculate the probability of rolling a sum of 5 at least 4 times in 5 throws, we need to consider two scenarios: rolling a sum of 5 four times and not rolling it once (k=4, n=5), and rolling a sum of 5 all five times (k=5, n=5).
Probability for exactly 4 successes:
(5 choose 4) * (1/9)^4 * (8/9)^1 = 5 * (1/6561) * (8/9) =~ 0.000072030
Probability for 5 successes:
(5 choose 5) * (1/9)^5 * (8/9)^0 = 1 * (1/59049) =~ 0.000016949
The total probability is the sum of these probabilities: approximately 0.000072030 + 0.000016949 = 0.000088979 or roughly 2/19683 when rounded to the nearest whole number fraction.