A set of$10$ cards consists of $5$ red cards and$5$ black cards. The cards are shuffled thoroughly, and you choose one at random, observe its color, and replace it in the set. The cards are thoroughly reshuffled, and you again choose a card at random, observe its color, and replace it in the set. This is done a total of four times. Let $X$ be the number of red cards observed in these four trials.

The random variable$X$ has which of the following probability distributions?

(a) The Normal distribution with mean$2$ and standard deviation$1$

(b) The binomial distribution with $n=10$androle="math" localid="1650534915843" $p=0.5$

(c) The binomial distribution with$n=5$ and $p=0.5$

(d) The binomial distribution with $n=4$and $p=0.5$

(e) The geometric distribution with $p=0.5$

We are given that a set of $10$cards consists of$5$red cards and 5 black cards, the cards are shuffled thoroughly, we choose one out of those at random, observe the color, and replace it into the set, cards are thoroughly reshuffled, again choose a card at random, observe its color, and replace it in the set, it is done a total of four times. Here $X$be the number of red cards observed in these four trials

We need to find that random variable $X$

We will find random variable $X$. For that it is given that cards are shuffled thoroughly, and you choose one at random, observe its color, and replace it in the set .Now we will find probability for different cards.

There are total of $10$cards out of which$5$are red and$5$are black ; so probability of red is $\frac{1}{2}=0.5$

,Similarly for black it is $0.5$.

Now cards are thoroughly reshuffled, again we will choose a card at random, observe its color, and replace it in the set, it is done a total of four times , so probability of red and black would be exact.

It is given that $X$be the number of red cards observed in these four trials , so this would be binomial experiment.

So, answer is binomial distribution with $n=4$and $p=0.5$