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.
P(head) + P(tail) = 1
P(head) = 3 P(tail)
Solving gives:
P(head) = 3/4
If we assign 1 to heads and a 0 to tail, then the mean value is 4P(head) = 3, the variance is 4P(head)(1-P(head)) =
3/4
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
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.
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To find the probability distribution of heads, we can consider each individual toss as a binary event where we can either get a head or a tail. Since a head is thrice as likely to occur as a tail, we can assign probabilities to each outcome.
Let's denote the probability of getting a head as P(head) and the probability of getting a tail as P(tail). We are given that a head is thrice as likely to occur as a tail, so we can write:
P(head) = 3 * P(tail)
We also know that the total probability of getting either a head or a tail must add up to 1, so we have:
P(head) + P(tail) = 1
We can solve these two equations to find the values of P(head) and P(tail).
Substituting the first equation into the second equation, we get:
3 * P(tail) + P(tail) = 1
Combining like terms, we have:
4 * P(tail) = 1
Dividing both sides by 4, we find:
P(tail) = 1/4
Using this value, we can now calculate the probability of getting a head:
P(head) = 3 * P(tail) = 3 * (1/4) = 3/4
So the probability distribution of getting a head is 3/4 and the probability distribution of getting a tail is 1/4.
To find the mean of the distribution, we can multiply the probability of getting a head by the number of tosses, which is 4 in this case:
Mean = 4 * P(head) = 4 * (3/4) = 3
So the mean of the distribution is 3.
To find the variance of the distribution, we can use the formula:
Variance = n * P(head) * (1 - P(head))
where n is the number of tosses. In this case, n = 4, P(head) = 3/4, so:
Variance = 4 * (3/4) * (1 - 3/4) = (12/4) * (1/4) = 3/4
So the variance of the distribution is 3/4.