A weighted coin is flipped 200 times. Assume that the probability of a head is 0.3

and the probability of a tail is 0.7. Each flip is independent from the other flips. Let
X be the total number of heads in the 200 flips.
1. What is the distribution of X ?
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2. What is the expected value of X and variance of X ?
3. What is the approximate probability that X equals 35 ?
4. What is the approximate probability that X is less than 45 ?

1. The distribution of X is a binomial distribution with parameters n = 200 (number of trials) and p = 0.3 (probability of success).

2. The expected value (mean) of X can be calculated using the formula: E[X] = n * p.
E[X] = 200 * 0.3 = 60.

The variance of X can be calculated using the formula: Var(X) = n * p * (1 - p).
Var(X) = 200 * 0.3 * (1 - 0.3) = 42.

3. To approximate the probability that X equals 35, we can use the normal approximation to the binomial distribution. Here, X follows a binomial distribution with parameters n = 200 and p = 0.3.

Using the normal approximation, we can calculate the mean (μ) and standard deviation (σ) of X.

μ = n * p = 200 * 0.3 = 60
σ = √(n * p * (1 - p)) = √(200 * 0.3 * 0.7) = 4.8989

Now we can calculate the z-score for X = 35:
z = (X - μ) / σ = (35 - 60) / 4.8989 ≈ -5.089

Using a standard normal distribution table or a calculator, we can find the probability corresponding to z = -5.089. The probability is very close to 0.

So, the approximate probability that X equals 35 is approximately 0.

4. To approximate the probability that X is less than 45, we can again use the normal approximation to the binomial distribution.

Using the same mean (μ) and standard deviation (σ) calculated above, we can calculate the z-score for X = 45:
z = (X - μ) / σ = (45 - 60) / 4.8989 ≈ -3.064

Using a standard normal distribution table or a calculator, we can find the probability corresponding to z = -3.064. The probability is approximately 0.0011.

So, the approximate probability that X is less than 45 is approximately 0.0011.

1. The distribution of X is a binomial distribution. In this case, since each flip is independent and the probability of success (getting a head) is fixed at 0.3, we can use the binomial distribution to describe the number of heads (X) in 200 flips.

2. The expected value of X is calculated as the product of the number of trials and the probability of success: E(X) = n * p. In this case, E(X) = 200 * 0.3 = 60.

To calculate the variance of X, we can use the formula var(X) = n * p * (1 - p). In this case, var(X) = 200 * 0.3 * (1 - 0.3) = 200 * 0.3 * 0.7 = 42.

3. To find the approximate probability that X equals 35, we can use the binomial probability formula. The probability of getting exactly k successes (heads) in n independent trials is given by P(X = k) = (n choose k) * p^k * (1 - p)^(n - k).
In this case, we have P(X = 35) = (200 choose 35) * 0.3^35 * 0.7^(200 - 35). Evaluating this expression using a calculator or statistical software will give you the approximate probability.

4. To find the approximate probability that X is less than 45, we can sum the probabilities of getting 0, 1, 2, ..., 44 successes. Using the binomial probability formula, we can calculate P(X < 45) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 44). Evaluating this expression using a calculator or statistical software will give you the approximate probability.