A six-sided die is rolled 36 times. What is the probability that an odd number is rolled at most 15 times?
What is the probability that an odd number is rolled exactly 15 times?
Use the Normal Approximation to a Binomial.
To find the probability that an odd number is rolled at most 15 times when a six-sided die is rolled 36 times, we can use the normal approximation to the binomial distribution.
First, let's define the values needed for the normal approximation:
n = 36 (number of trials)
p = 1/2 (probability of rolling an odd number)
q = 1 - p = 1/2 (probability of rolling an even number)
μ = n * p = 36 * 1/2 = 18 (mean)
σ = √(n * p * q) = √(36 * 1/2 * 1/2) = √(9) = 3 (standard deviation)
To find the probability that an odd number is rolled at most 15 times, we need to find the cumulative probability up to 15.
Using the normal approximation, we can use the z-score formula:
z = (x - μ) / σ,
where x is the number of successes (15 in this case).
For P(X ≤ 15):
z = (15 - 18) / 3 = -1
Next, we can look up the z-score in the standard normal distribution table or use a calculator to find the corresponding probability.
Using a standard normal distribution table, the probability corresponding to a z-score of -1 is approximately 0.1587.
Thus, the probability that an odd number is rolled at most 15 times is approximately 0.1587.
To find the probability that an odd number is rolled exactly 15 times, we need to find P(X = 15).
Since the binomial distribution is discrete, we need to calculate the difference between the cumulative probability at 15 (P(X ≤ 15)) and the cumulative probability at 14 (P(X ≤ 14)).
Using the normal approximation, we calculate the z-scores for 14 and 15:
z_14 = (14 - 18) / 3 = -4/3
z_15 = (15 - 18) / 3 = -1
Next, we find the probabilities corresponding to these z-scores in the standard normal distribution table.
P(X = 15) = P(X ≤ 15) - P(X ≤ 14)
= P(z ≤ -1) - P(z ≤ -4/3)
Using the standard normal distribution table or a calculator, we find that the probabilities corresponding to z = -1 and z = -4/3 are approximately 0.1587 and 0.0912 respectively.
P(X = 15) = 0.1587 - 0.0912 ≈ 0.0675
Thus, the probability that an odd number is rolled exactly 15 times is approximately 0.0675.
To find the probability that an odd number is rolled at most 15 times when a six-sided die is rolled 36 times, we can use the Normal Approximation to a Binomial.
Step 1: Find the mean and standard deviation of the binomial distribution.
The mean (μ) of a binomial distribution is given by the formula: μ = n * p, where n is the number of trials and p is the probability of success in a single trial. In this case, n = 36 (number of rolls) and p = 1/2 (probability of rolling an odd number). So μ = 36 * 1/2 = 18.
The standard deviation (σ) of a binomial distribution is given by the formula: σ = √(n * p * (1 - p)). So σ = √(36 * 1/2 * (1 - 1/2)) = √(36 * 1/2 * 1/2) = √(9) = 3.
Step 2: Convert the problem into a normal distribution problem.
To use the Normal Approximation to a Binomial, we need to transform our problem into a normal distribution problem.
For "at most 15 times", we need to find the probability that the number of odd rolls is less than or equal to 15.
We can calculate the z-score using the formula: z = (x - μ) / σ, where x is the number of successes, μ is the mean, and σ is the standard deviation. In this case, x = 15, μ = 18, and σ = 3. So z = (15 - 18) / 3 = -1.
Step 3: Find the probability using the standard normal distribution table.
Using the standard normal distribution table, we can find the probability associated with the z-score of -1. We can either use a table or a calculator to find this probability.
Looking up the z-score of -1 in the standard normal distribution table, we find the probability to be approximately 0.1587.
Therefore, the probability that an odd number is rolled at most 15 times is approximately 0.1587.
To find the probability that an odd number is rolled exactly 15 times, you can subtract the probability of 15 or fewer odd rolls from the probability of 14 or fewer odd rolls.
P(exactly 15 odd rolls) = P(15 or fewer odd rolls) - P(14 or fewer odd rolls)
P(exactly 15 odd rolls) = P(odd rolls ≤ 15) - P(odd rolls ≤ 14)
You can use the same method as above to calculate these probabilities.