An unfair coin is weighted so that the expected number of tails before the first head is 3.what is the standard deviation of the number of tails that come up before the first head
To find the standard deviation of the number of tails that come up before the first head, we need to understand the distribution of this random variable.
Let's denote the random variable as X, representing the number of tails that come up before the first head. According to the problem, the expected value (mean) of X is given as 3.
Now, for a coin toss, the probability of getting a head is p, and the probability of getting a tail is 1 - p. Since it is an unfair coin, the probabilities of getting a head and a tail are not necessarily 1/2 each.
To calculate the probability distribution of X, we can use the geometric distribution. The probability mass function (PMF) of the geometric distribution for X is given by:
P(X = x) = (1 - p)^(x) * p
Where x represents the number of tails before the first head. In this case, we want to find the standard deviation, which depends on the variance of X.
The variance of X for a geometric distribution is given by:
Var(X) = (1 - p) / (p^2)
To find the variance, we need to find the value of p. Given the mean (expected value) of X is 3, we can use this information to find p.
The expected value of a geometric distribution is given by:
E(X) = (1 - p) / p
Since E(X) = 3, we can set up the equation:
3 = (1 - p) / p
Solving this equation, we get:
3p = 1 - p
4p = 1
p = 1/4
Now that we have the value of p, we can calculate the variance:
Var(X) = (1 - p) / (p^2)
= (1 - 1/4) / ((1/4)^2)
= (3/4) / (1/16)
= (3/4) * (16/1)
= 12
Finally, to find the standard deviation, we take the square root of the variance:
Standard Deviation = sqrt(Var(X))
= sqrt(12)
= 2*sqrt(3)
Therefore, the standard deviation of the number of tails that come up before the first head is 2*sqrt(3).