Question

Consider the situation where you have a bag that contains 8 black marbles and 2 white marbles. You take a marble from the bag; you record whether it is black or white, and put it back in the bag before you take another marble from the bag. You do this 10 times. What is the probability that you will draw the same number of black and white marble?

Explain and show step by step process to achieve full marks.

Answer 1

Here, the scenario can be compared to that of SRSWR(10).
There are 8 black marbles and 2 white marbles in a bag.
Therefore, total number of marbles = 8+2 =10.

Therefore, chance of picking a white marble in a single turn is
= choosing a white marble out of 8 white marbles/choosing a marble out of 10 marbles


Therefore, chance of picking a black marble in a single turn is
= choosing a black marble out of 8 white marbles/choosing a marble out of 10 marbles


Let X be a random variable denoting the number of white marbles picked in 10 turns.

Clearly,

[Note: If X white marbles are picked, 10 - X black marbles are picked.
Now, same number of white and black marbles are picked.
i.e.,


]

The p.m.f. of X is given by:
,



.

Therefore, the probability that you will draw the same number of black and white marble is 0.0264241152.


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