In a binomial distribution the mean is 3 and standard deviation is 1.5, find the parameters of binomial distribution.
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The parameters of a binomial distribution are the number of trials (n) and the probability of success (p). In this case, n = 3 and p = 1.5.
In a random experiment of rolling a die, let A be the event that an even number occurs and B be the event that a multiple of 3 occurs. Find P(A) and P(B). Is A and B equally likely events?
To find the parameters of a binomial distribution when given the mean and standard deviation, you can use the formulas:
Mean (μ) = n * p
Standard Deviation (σ) = √(n * p * (1 - p))
where:
- n is the number of trials
- p is the probability of success in each trial
In this case, you are given that the mean (μ) is 3 and the standard deviation (σ) is 1.5.
Step 1: Use the formula for the mean to find the value of n * p.
3 = n * p
Step 2: Rearrange the formula to solve for either n or p.
p = 3 / n
Step 3: Substitute this value of p into the formula for the standard deviation and solve for n.
1.5 = √(n * (3 / n) * (1 - (3 / n)))
1.5 = √(3 * (1 - (3 / n)))
Squaring both sides of the equation:
2.25 = 3 * (1 - (3 / n))
Distributing and simplifying:
2.25 = 3 - (9 / n)
(9 / n) = 3 - 2.25
(9 / n) = 0.75
Cross-multiplying:
9 = 0.75n
Dividing both sides by 0.75:
n = 9 / 0.75
n = 12
Step 4: Substitute the value of n into the formula for p to find its value.
p = 3 / 12
p = 0.25
Therefore, the parameters of the binomial distribution are n = 12 and p = 0.25.