Assume that the coin is weighted so that Pr[H] = [ 1/11], and that it is flipped 9 times. The random variable X is defined as the number of heads in the 9 flips.
What is E[X]?
To find the expected value, E[X], we can use the formula:
E[X] = (probability of outcome 1 * value of outcome 1) + (probability of outcome 2 * value of outcome 2) + ... + (probability of outcome n * value of outcome n)
In this case, the possible values of X are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The probability of each outcome depends on the weighted coin.
Since the probability of getting a head (H) is 1/11, the probability of getting a tail (T) is 10/11.
To calculate E[X], we need to consider the probabilities for each possible outcome 0, 1, 2, ..., 9.
To get 0 heads, the probability is (10/11) * (10/11) * ... * (10/11) (nine times), which is (10/11)^9.
To get 1 head, the probability is (1/11) * (10/11)^8 (one head and eight tails in any order).
To get 2 heads, the probability is C(9, 2) * (1/11)^2 * (10/11)^7 (using the binomial coefficient to choose 2 positions for the heads).
Similarly, for 3 heads, the probability is C(9, 3) * (1/11)^3 * (10/11)^6, and so on.
Now, we can calculate the expected value E[X].
E[X] = (0 * (10/11)^9) + (1 * (1/11) * (10/11)^8) + (2 * C(9, 2) * (1/11)^2 * (10/11)^7) + ... + (9 * C(9, 9) * (1/11)^9)
= 0 + (1 * (1/11) * (10/11)^8) + (2 * C(9, 2) * (1/11)^2 * (10/11)^7) + ... + (9 * C(9, 9) * (1/11)^9)
You can calculate each term above and sum them up to find the expected value E[X].