Q.A container contains 70 l of orange squash. The squash being too concentrated 7 l of squash was taken out from this container and replaced by water. This process was repeated thrice to reduce the concentration of squash. How much quantity of orange squash is left in the container?

Successive Replacement

Imagine you have a bag of 10 marbles — 4 red and 6 blue. You draw one marble, note its colour, and put it back. Then you draw another. The chance that both draws are red is $\frac{4}{10} \times \frac{4}{10} = 0.16$. That's straightforward because the bag is exactly the same for the second draw.

Now change the game: you draw a marble and do not put it back. The bag now has only 9 marbles. If the first was red, only 3 reds remain; if it was blue, all 4 reds are still there. The probability of the second event depends on what happened first. This is the core of successive replacement — or rather, its absence.

Successive replacement is the process of drawing items one after another from a finite set, returning each item to the set before the next draw. The key consequence: the composition of the set never changes. Every draw is identical and independent of every other draw.

The Precise Statement

Let a set contain $N$ items, of which $K$ are of a particular type (say "success"). You draw $n$ items one at a time, replacing each item before the next draw. Then:

• The probability of getting a success on any single draw is always $\frac{K}{N}$.
• The draws are independent — the outcome of one draw does not affect the next.
• The number of successes in $n$ draws follows a binomial distribution:

Why It Matters

Without replacement, probabilities shift after every draw — that's the hypergeometric distribution, and it's messier. With replacement, you get the clean, constant-probability binomial model. This is why successive replacement is the default assumption in many exam problems unless "without replacement" is explicitly stated.

Common exam trap: A problem says "three balls are drawn from a bag containing 5 red and 7 green balls." If it does not say "without replacement," check the context. In probability, "drawn" alone usually means with replacement unless specified otherwise. In combinatorics (counting arrangements), "drawn" often means without replacement. Read carefully.

Quick Example

A box has 3 defective bulbs and 7 good bulbs. You pick a bulb, test it, and put it back. You do this 4 times. What is the probability that exactly 2 of the 4 picks are defective?

Here $N=10$, $K=3$, $p = \frac{3}{10} = 0.3$, $n=4$, $r=2$: