4. A balanced coin is tossed 10 times. Let X denote the number of heads. Calculate the following:
4.1. P(X = 4)
4.2. P(X 1)
4.3. P(X > 1)
4.4. P(3< X 7)
4.5. E(X)
4.6. Var(X)
To calculate the probabilities, we need to use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- n is the number of trials (10 in this case)
- k is the number of successes (heads) we want to find the probability for
- p is the probability of success on a single trial (0.5 for a balanced coin toss)
- (n choose k) is the binomial coefficient, which can be calculated as n! / (k! * (n-k)!)
We will use this formula to calculate the probabilities:
4.1. P(X = 4)
P(X = 4) = (10 choose 4) * (0.5)^4 * (0.5)^(10-4)
P(X = 4) = 210 * 0.0625 * 0.0625
P(X = 4) = 0.2051
4.2. P(X > 1)
P(X > 1) = 1 - P(X = 0) = 1 - (10 choose 0) * (0.5)^0 * (0.5)^10
P(X > 1) = 1 - 1 * 1 * 0.00098
P(X > 1) = 0.999
4.3. P(X > 1) = 0.999
4.4. P(3 < X < 7)
P(3 < X < 7) = P(X = 4) + P(X = 5) + P(X = 6)
P(3 < X < 7) = 0.2051 + 0.2461 + 0.2051
P(3 < X < 7) = 0.6563
4.5. E(X) is calculated as n * p:
E(X) = 10 * 0.5
E(X) = 5
4.6. Var(X) is calculated as n * p * (1-p):
Var(X) = 10 * 0.5 * 0.5
Var(X) = 2.5
These are the calculations for the probabilities and expected value and variance of X in this scenario.