Calculate the probability that 200 rolls of
two dice rolls will include
a) more than 30 sums of 5
b) between 30 and 40, inclusive, sums of 5
To calculate the probability, we first need to determine the total number of possible outcomes for each question.
a) More than 30 sums of 5:
In order to have more than 30 sums of 5, we need to calculate the number of ways to roll a sum of 5 using two dice. Then, we can determine the probability of rolling a sum of 5 on each individual roll.
Step 1: Calculate the number of ways to roll a sum of 5:
The possible combinations that result in a sum of 5 are: (1, 4), (2, 3), (3, 2), and (4, 1). So, there are 4 possible ways to roll a sum of 5.
Step 2: Determine the probability of rolling a sum of 5 on each roll:
There are 36 possible outcomes when rolling two dice (6 sides on each die). The probability of rolling a sum of 5 is the number of favorable outcomes (4) divided by the total number of possible outcomes (36), which gives us 4/36 or 1/9.
Step 3: Calculate the probability of rolling more than 30 sums of 5:
To calculate this probability, we need to use the binomial distribution formula. The formula for the probability of getting x successes in n trials, where the probability of success is p, is given by:
P(X = x) = (nCx) * (p^x) * ((1 - p)^(n-x))
In this case, we want to calculate the probability of getting more than 30 sums of 5, which means we need to find the sum of probabilities for getting 31, 32, ..., 200 sums of 5.
Using a calculator or a programming language, we can calculate the probability for each value of x and sum them up to get the final probability.
b) Between 30 and 40, inclusive, sums of 5:
To calculate this probability, we follow the same steps as in part a, but instead of summing up the probabilities for values of x from 31 to 200, we sum up the probabilities for values of x from 30 to 40 (inclusive).
Again, using a calculator or a programming language, we can calculate the probability for each value of x and sum them up.
Note: The calculations involved can be complex, so it's recommended to use a programming language or a calculator to perform the necessary calculations.
To calculate the probability for these scenarios, we need to understand the probability distribution of the sum of two dice rolls.
First, let's determine the number of possible outcomes from rolling two dice. Each die has six faces, so the total number of outcomes is 6 * 6 = 36.
Next, let's determine the number of ways to get a sum of 5 from rolling two dice. There are four combinations that can result in a sum of 5: (1, 4), (4, 1), (2, 3), and (3, 2).
Now, let's move on to calculating the probabilities:
a) More than 30 sums of 5:
To calculate the probability of getting more than 30 sums of 5, we can use the binomial distribution formula. The formula is as follows:
P(X > k) = 1 - P(X ≤ k)
Where:
P(X > k) is the probability of getting more than k successes (in this case, sums of 5)
P(X ≤ k) is the probability of getting k or fewer successes (in this case, sums of 5)
Using the formula mentioned above, we can calculate the probability as follows:
P(X > 30) = 1 - P(X ≤ 30)
To calculate P(X ≤ 30), we need to find the probability of getting 0, 1, 2, ..., 30 sums of 5 and sum them up.
Let's calculate the probability of getting exactly k sums of 5 using the binomial distribution formula:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
Where:
n is the number of rolls (200 in this case)
k is the number of successes (sums of 5)
p is the probability of getting a success (probability of rolling a sum of 5)
Since there are four combinations that can result in a sum of 5 and a total of 36 possible outcomes, the probability of getting a sum of 5 is p = 4/36 = 1/9.
Using these values, we can calculate P(X ≤ 30) by summing up the probabilities of getting 0 to 30 sums of 5:
P(X ≤ 30) = P(X = 0) + P(X = 1) + ... + P(X = 30)
After obtaining P(X ≤ 30), we can calculate P(X > 30) as 1 - P(X ≤ 30).
b) Between 30 and 40, inclusive, sums of 5:
To calculate the probability of getting between 30 and 40 (inclusive) sums of 5, we need to sum the probabilities of getting 30, 31, 32, ..., 40 sums of 5:
P(30 ≤ X ≤ 40) = P(X = 30) + P(X = 31) + ... + P(X = 40)
By calculating P(X ≤ 30) and P(30 ≤ X ≤ 40) separately, we can calculate the probability between 30 and 40 sums of 5 as P(30 ≤ X ≤ 40) - P(X ≤ 30).
Using these calculations, we can find the probabilities for both scenarios.