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)
4.1. To calculate P(X = 4), we can use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
where
n = total number of trials = 10
k = number of heads = 4
p = probability of heads on a single trial = 0.5 (since the coin is balanced)
Plugging in the values, we get:
P(X = 4) = (10 choose 4) * (0.5)^4 * (0.5)^(10-4)
= 210 * 0.0625 * 0.0625
= 0.2051
Therefore, P(X = 4) = 0.2051
4.2. To calculate P(X < 1), it means finding the probability of getting 0 heads (since getting 0 heads is the only event < 1).
P(X < 1) = P(X = 0) = (10 choose 0) * (0.5)^0 * (0.5)^(10-0)
= 1 * 1 * 0.0009765625
= 0.0009765625
Therefore, P(X < 1) = 0.0009765625
4.3. To calculate P(X > 1), it means finding the probability of getting more than 1 head. This is the complement of getting either 0 or 1 head.
P(X > 1) = 1 - P(X < 1)
= 1 - P(X = 0)
= 1 - 0.0009765625
= 0.9990234375
Therefore, P(X > 1) = 0.9990234375