Use of normal distribution to approximate the desired probability. Find the probability of getting at least 30 fives in 200 tosses of a fair 6 sided die.
To find the probability of getting at least 30 fives in 200 tosses of a fair 6-sided die using the normal distribution, we can make use of the Central Limit Theorem.
The Central Limit Theorem states that, under certain conditions, the sum or average of a large number of independent and identically distributed random variables will be approximately normally distributed, regardless of the shape of the original distribution.
Here's how we can apply the Central Limit Theorem to solve this problem step-by-step:
1. Define the random variable:
Let X be the number of fives obtained in 200 tosses of the die.
2. Calculate the mean and standard deviation:
For a fair 6-sided die, the probability of getting a five on a single toss is 1/6. Therefore, the mean (μ) of X is given by:
μ = (1/6) * 200 = 33.33
The variance (σ²) of X is given by:
σ² = (1/6) * (5/6) * 200 = 55.56
The standard deviation (σ) of X is the square root of the variance:
σ = √55.56 ≈ 7.46
3. Approximate with the normal distribution:
Since the number of tosses is large (200), we can approximate the distribution of X with a normal distribution with mean μ = 33.33 and standard deviation σ = 7.46.
4. Standardize the desired value:
We want to find the probability of getting at least 30 fives, which can be written as P(X ≥ 30). To use the normal distribution, we need to standardize this value.
The standardized value (Z-score) is calculated as:
Z = (X - μ) / σ
For X = 30, the Z-score is:
Z = (30 - 33.33) / 7.46 ≈ -0.444
5. Calculate the probability using the standard normal distribution:
To find the probability of getting at least 30 fives, we need to calculate P(X ≥ 30), which is equivalent to finding P(Z ≥ -0.444).
Using a standard normal distribution table or a calculator, we can find that P(Z ≥ -0.444) is approximately 0.657.
Therefore, the probability of getting at least 30 fives in 200 tosses of a fair 6-sided die is approximately 0.657, or 65.7%.