Animated Solution for Mathematics - Probability: From a lot containing 10 defective and 90 non-defective bulbs, 8 bulbs are selected one by one with replacement. Then the probability of getting at least 7 defective bulbs is :

Imagine standing before a large box containing exactly one hundred bulbs. We are told that ten are defective, and ninety are perfectly fine. We are tasked with selecting eight bulbs, one by one, with replacement.

The "with replacement" clause is the heartbeat of this problem. It ensures that every time we pick a bulb, we note its status and return it to the box, keeping the probability of picking a defective bulb constant at $p=\frac{10}{100}=0.1$ for every single draw.

This independence is what allows us to use the Binomial Distribution.

In the language of probability, we define 'success' as picking a defective bulb. Thus, $p=0.1$. Consequently, 'failure' is picking a non-defective bulb, with probability $q=1-p=0.9$.

We are performing $n=8$ trials. The Binomial Distribution formula is our map:

This formula provides the probability of getting exactly $r$ successes in $n$ trials.

The question asks for the probability of getting 'at least 7' defective bulbs. This means we are interested in the scenarios where we get exactly 7 defective bulbs OR exactly 8 defective bulbs.

Mathematically, we need to calculate:

For $r=7$, we calculate:

Since $\left(\genfrac{}{}{0ex}{}{8}{7}\right)=8$, this becomes:

For $r=8$, we calculate:

Since $\left(\genfrac{}{}{0ex}{}{8}{8}\right)=1$ and $(0.9{)}^0=1$, this simplifies to:

Finally, we sum these probabilities to find the total likelihood:

The final probability is $7.3\times 10^{-7}$ (or $0.00000073$). This result is the culmination of understanding independence, defining success, and applying the Binomial model.