Question

Find the probability that event A will appear at least three times in four independent tests if the probability of occurrence of event A in one test equal to 0.4

Answer 1

This kind of problems are solved using Bernoulli's distribution. Everytime you have a win/lose scenario, and you know the probability $p$ of winning, and you want $k$ successes over $n$ trials, you have the following probability:

$\displaystyle P(k\text{ successes over }n\text{ trials}) = \binom{n}{k}p^k(1-p)^{n-k}$

You want the probability of having at least three successes, i.e. you are interested in the cases k=3 and k=4. The corresponding probabilities are

$\displaystyle P(3 \text{ successes}) = \binom{4}{3}0.6^3 \cdot 0.4 = 4\cdot 0.216 \cdot 0.4 = 0.3456$

$\displaystyle P(4 \text{ successes}) = \binom{4}{4}0.6^4 = 0.1296$

So, the total probability is $0.3456+0.1296 = 0.4752$