Question

A set of 10 cards consists of five red cards and five black cards. The cards are shuffled thoroughly, and I choose one at random, observe its color, and replace it in the set. The cards are thoroughly reshuffled, and I 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 mean of X is:

Answer 1

To find the mean of X, we need to calculate the average number of red cards observed in the four trials. The mean of X is 0.5.

Step-by-step explanation:

To find the mean of X, we need to calculate the average number of red cards observed in the four trials. In each trial, there is a 50% chance of selecting a red card and a 50% chance of selecting a black card. Since the cards are replaced after each trial, the probability remains the same.

So, the probability of selecting a red card in each trial is 0.5. Since there are four trials, we can calculate the mean as:

Mean = (0.5 + 0.5 + 0.5 + 0.5) / 4 = 2 / 4 = 0.5.

Therefore, the mean of X is 0.5.

Answer 2

The mean of X is 2

Step-by-step explanation:

Given the data in the question;

set of 10 cards consist of 5 red card and 5 blacks

cards are chosen, replaced and shuffled four times;

number of independent trials n = 4

now, since its is replaced, the probability of getting a red card in each trial will be;

p = 5/10 = 0.5

let X rep the number of red cards observed in the four trials.

the random variable follows binomial distribution where

n = 4 and p = 0.5

so

E(X) = ∑xP( X = x )

E(X) = np

we substitute

E(X) = 4 × 0.5

E(X) = 2

Therefore, The mean of X is 2