The formula to calculate the probability of success after N trials is:
\[ P(X=k) = C(n, k) \cdot p^k \cdot (1-p)^{(n-k)} \]
Where:
The Probability of Success After N Trials, often referred to in statistics as the binomial probability, is the likelihood of achieving a specific number of "successes" in a set number of trials or attempts. A "success" is defined as the outcome that we are interested in. This concept is used in binomial experiments where each trial is independent and has only two possible outcomes - success or failure. The probability remains constant throughout all trials.
Let's assume the following values:
Step 1: Calculate the combinations:
\[ C(10, 3) = \frac{10!}{3!(10-3)!} = 120 \]
Step 2: Calculate the probability of k successes:
\[ P(X=3) = 120 \cdot 0.5^3 \cdot (1-0.5)^{(10-3)} = 0.1172 \]
Therefore, the probability of having exactly 3 successes out of 10 trials is approximately 0.1172.
