A coin is biased such that a head is thrice as likely to occur as a tail. Find the probability distribution of heads and also find the mean and variance of the distribution when it is tossed 4 times.
See http://www.stat.yale.edu/Courses/1997-98/101/binom.htm
Let p be the probablility of heads = 0.75
Nor N = 4 tosses, the mean number is N*p = 3 and the variance is
sigma^2 =
p(1-p)N = (1/4)*(3/4)*4 = 3/4
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To find the probability distribution of heads when a biased coin is tossed four times, we need to consider the probability of obtaining each possible number of heads.
Let's denote the probability of getting heads as p, and the probability of getting tails as q (which is simply 1 minus the probability of getting heads). According to the problem, the probability of getting heads is three times more likely than getting tails, so we have p = 3q.
To find the probability distribution, we can use the binomial probability formula. The probability of obtaining k heads in n tosses is given by:
P(X = k) = C(n, k) * p^k * q^(n-k)
Where C(n, k) represents the number of combinations of n objects taken k at a time. In this case, n = 4.
Let's calculate the probability distribution for each possible number of heads:
P(X = 0) = C(4, 0) * (3/4)^0 * (1/4)^4 = 1 * 1 * 1/256 = 1/256
P(X = 1) = C(4, 1) * (3/4)^1 * (1/4)^3 = 4 * 3/4 * 1/64 = 12/256 = 3/64
P(X = 2) = C(4, 2) * (3/4)^2 * (1/4)^2 = 6 * 9/16 * 1/16 = 54/256 = 27/128
P(X = 3) = C(4, 3) * (3/4)^3 * (1/4)^1 = 4 * 27/64 * 1/4 = 108/256 = 27/64
P(X = 4) = C(4, 4) * (3/4)^4 * (1/4)^0 = 1 * 81/256 * 1 = 81/256
So, the probability distribution of heads for tossing the biased coin four times is as follows:
P(X = 0) = 1/256
P(X = 1) = 3/64
P(X = 2) = 27/128
P(X = 3) = 27/64
P(X = 4) = 81/256
To find the mean (expected value) of this distribution, we can use the formula:
Mean = Σ(X * P(X))
Mean = 0 * (1/256) + 1 * (3/64) + 2 * (27/128) + 3 * (27/64) + 4 * (81/256)
Mean = 0 + 3/64 + 54/128 + 81/64 + 324/256
Mean = (3 + 54 + 162 + 324) / 256
Mean = 543/256
The mean of the probability distribution is 543/256.
To find the variance, we can use the formula:
Variance = Σ((X - Mean)^2 * P(X))
Variance = (0 - 543/256)^2 * (1/256) + (1 - 543/256)^2 * (3/64) +
(2 - 543/256)^2 * (27/128) + (3 - 543/256)^2 * (27/64) + (4 - 543/256)^2 * (81/256)
Calculating this expression will give us the variance.
Variance = 649347/262144
The variance of the probability distribution is 649347/262144.
(Note: Some rounding may be necessary depending on the level of precision required.)