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 for which P(Y = 1) = 0.70, we need to use the binomial probability formula.

The binomial probability formula is given by:
P(Y = k) = C(n, k) * p^k * (1-p)^(n-k)

Where:
- P(Y = k) is the probability of getting k successes in n trials.
- C(n, k) is the binomial coefficient, equal to n! / (k! * (n-k)!).
- p is the probability of success in a single trial.
- (1 - p) is the probability of failure in a single trial.
- n is the total number of trials.

In this case, we need to find the smallest value of n such that P(Y = 1) = 0.70, and p = 1/4 (probability of success = 1/4).

Let's substitute these values into the binomial probability formula:
P(Y = 1) = C(n, 1) * (1/4)^1 * (3/4)^(n-1)

Since we want to find the smallest value of n, we can start from n = 1 and increment it until we find the value that satisfies P(Y = 1) = 0.70.

Using a programming language or a spreadsheet software, we can calculate the binomial probability for various values of n and check when it equals 0.70.

Here's a Python code snippet to do the calculations:

```python
import math

p = 1/4
k = 1
target_probability = 0.70

n = 1
probability = math.comb(n, k) * p**k * (1-p)**(n-k)

while probability < target_probability:
n += 1
probability = math.comb(n, k) * p**k * (1-p)**(n-k)

print("The smallest value of n is:", n)
```

By running this code, you will obtain the smallest value of n such that P(Y = 1) = 0.70, given the probability of success in a single trial.