a set of 10 cards consists of 5 red cards and 5 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 thouruoghly 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?
Four trials x 50% probability of a red each time = 2
To find the mean value of x, we need to determine the probability of observing a red card in each trial, and then calculate the expected value.
Given that there are 5 red and 5 black cards in the set of 10:
1. In the first trial, the probability of observing a red card is 5/10 or 1/2.
2. After each trial, the cards are replaced and the set is thoroughly reshuffled. So, the probability of observing a red card remains the same for each subsequent trial.
Since we are conducting 4 trials, we can consider each trial as an independent event.
Now, let's calculate the mean of x:
Mean = (Sum of all possible values of x) / (Total number of trials)
Possible values of x: 0, 1, 2, 3, 4
1. For x = 0: The probability of observing zero red cards in all four trials is (1/2)^4.
2. For x = 1: The probability of observing one red card in any one of the four trials is 4 * (1/2)^4 because there are four different trials where you can observe a red card.
3. For x = 2: The probability of observing two red cards in two out of four trials is (4 choose 2) * (1/2)^4.
4. For x = 3: The probability of observing three red cards in three out of four trials is (4 choose 3) * (1/2)^4.
5. For x = 4: The probability of observing four red cards in all four trials is (1/2)^4.
Summing up the probabilities for each value of x:
Mean = [0 * (1/2)^4 + 1 * 4 * (1/2)^4 + 2 * (4 choose 2) * (1/2)^4 + 3 * (4 choose 3) * (1/2)^4 + 4 * (1/2)^4] / 4
Simplifying this expression gives us:
Mean = (0 + 4/16 + 6/16 + 4/16 + 1/16) / 4
= (15/16) / 4
= 15/64
Therefore, the mean value of x is 15/64.
To calculate the mean of x, we need to determine the probability of observing a red card in a single trial and then use that probability to find the expected value.
In each trial, you have a 5 out of 10 chance of selecting a red card because there are 5 red cards out of a total of 10 cards. Therefore, the probability of selecting a red card in a single trial is 5/10 or 1/2.
Since the card is replaced after each selection, the probability of selecting a red card remains the same for each trial. Therefore, the trials are independent events.
The number of red cards observed, x, follows a binomial distribution because each trial has two possible outcomes (red or black) and the trials are independent. The expected value or mean of a binomial distribution is given by the product of the number of trials (4 in this case) and the probability of success in each trial (1/2 for red card selection).
Mean (Expected value) = Number of trials x Probability of success in each trial
Mean (Expected value) = 4 x (1/2) = 2
Therefore, the mean of x, the number of red cards observed in the four trials, is 2.