Answer:
See below
Step-by-step explanation:
To find the probability of throwing a sum of 5 at least 8 times in 11 throws of a pair of fair dice, we can use the concept of binomial probability.
First, let's determine the probability of getting a sum of 5 in a single throw. To get a sum of 5, we can have the following combinations: (1, 4), (2, 3), (3, 2), and (4, 1). Each combination has a probability of 1/36 (1 out of 36 possible outcomes) since there are 36 possible outcomes when rolling a pair of fair dice.
Now, we need to calculate the probability of getting a sum of 5 at least 8 times in 11 throws. To do this, we'll use the binomial probability formula:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of getting exactly k successes,
- n is the total number of trials,
- k is the number of desired successes,
- p is the probability of success in a single trial,
- (n C k) is the binomial coefficient, calculated as n! / (k! * (n - k)!)
In our case, n = 11 (the total number of throws), k = 8 (the desired number of times to get a sum of 5), and p = 1/36 (the probability of getting a sum of 5 in a single throw).
Now, let's calculate the probability using the formula:
P(X >= 8) = P(X = 8) + P(X = 9) + ... + P(X = 11)
P(X = 8) = (11 C 8) * (1/36)^8 * (35/36)^(11 - 8)
P(X = 9) = (11 C 9) * (1/36)^9 * (35/36)^(11 - 9)
P(X = 10) = (11 C 10) * (1/36)^10 * (35/36)^(11 - 10)
P(X = 11) = (11 C 11) * (1/36)^11 * (35/36)^(11 - 11)
Let's calculate each probability:
P(X = 8) = (11 C 8) * (1/36)^8 * (35/36)^3
P(X = 9) = (11 C 9) * (1/36)^9 * (35/36)^2
P(X = 10) = (11 C 10) * (1/36)^10 * (35/36)^1
P(X = 11) = (11 C 11) * (1/36)^11 * (35/36)^0
After calculating each probability, we can add them up to find the final probability:
P(X >= 8) = P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11)
Keep in mind that the binomial coefficient (n C k) represents the number of ways to choose k successes from n trials, and it can be calculated as n! / (k! * (n - k)!).
I hope this explanation helps you understand how to find the probability of throwing a sum of 5 at least 8 times in 11 throws of a pair of fair dice using the binomial probability formula. If you have any further questions, feel free to ask.