Question

The probability of success in an experiment is p = 0.1. If the experiment is conducted 7 times, what is the probability that 3 or less will end in success?

Answer 1

Answer on Question #44150 – Math - Statistics and Probability

The probability of success in an experiment is $p = 0.1$. If the experiment is conducted 7 times, what is the probability that 3 or less will end in success?

Solution

Assume that trials are independent. Let the random variable $X$ denote the number of successes among 7 trials. Hence $X$ has a binomial distribution with $n = 7$ and $p = 0.1$.

We calculate probability

$$\begin{aligned} P(X \leq 3) &= P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = \\ &= \binom{7}{0} (0.1)^0 (0.9)^7 + \binom{7}{1} (0.1)^1 (0.9)^6 + \binom{7}{2} (0.1)^2 (0.9)^5 + \binom{7}{3} (0.1)^3 (0.9)^4 = \\ &= (0.9)^7 + 7 * 0.1(0.9)^6 + 21(0.1)^2(0.9)^5 + 35(0.1)^3(0.9)^4 \approx 0.997, \end{aligned}$$

where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, $n! = 1 \times 2 \times \ldots \times n$.

Answer: 0.997.

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