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 $n=10$and $p=0.5$

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$p=0.5$

Step by step solution

01

Given information

A set of $10$cards consists of $5$ red cards and $5$ black cards.

02

Explanation 

There are ten cards in total, five red and five black. It is also assumed that the cards are shuffled and that one is chosen at random.

Red has a $0.5$probability, while black has a $0.5$probability (since$5$out of the$10$cards are red and black).

The important part is that the card is replaced and shuffled afresh, ensuring that the next draw is unaffected by the previous one.

This is done four times with the same exact chance of receiving red and black each time.

This would be a binomial experiment if$X$ is a number of red cards. As a result, option (d) is the proper choice.

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