Let Y be the number of successes in n independent repetitions of a random experiment have
probability of success = 1/4. Determine the smallest value of n so that P(1 = Y) = 0.70
To determine the smallest value of n so that P(Y = 1) = 0.70, where Y represents the number of successes in n independent repetitions of a random experiment with a probability of success of 1/4, we can use the binomial probability formula.
The binomial probability formula states that the probability of getting exactly k successes in n independent repetitions of an experiment, where the probability of success in each repetition is p, is given by:
P(Y = k) = nCk * p^k * (1 - p)^(n - k)
where nCk represents the number of combinations (also known as binomial coefficient) of choosing k successes from n trials, which can be calculated using the formula:
nCk = n! / [k! * (n - k)!]
In our case, we want to find the smallest value of n such that P(Y = 1) = 0.70. So we need to solve the equation:
P(Y = 1) = nC1 * (1/4)^1 * (3/4)^(n - 1) = 0.70
Let's start by calculating the value of nC1:
nC1 = n! / [1! * (n - 1)!] = n
Now, we can substitute these values back into the equation:
n * (1/4) * (3/4)^(n - 1) = 0.70
To solve this equation, we can use a numerical method, such as trial and error or using a software tool like a graphing calculator or solver.
By trying different values of n, we find that the smallest value of n that satisfies the equation is approximately 11.