Question

A small deck of 5 cards consists of 3 red cards and 2 green cards. Draw 15 times with replacement, where each draw is random. Let X be the number of red cards drawn. Use the normal distribution to approximate P(X < 7).

Answer 1

To approximate P(X < 7) in this problem, we can use the normal distribution. The approximate probability that X is less than 7 is about 18.07%.

Step-by-step explanation:

In this problem, we have a small deck of 5 cards with 3 red cards and 2 green cards. We are drawing 15 times with replacement, meaning that after each draw, the card is placed back into the deck. Let X be the number of red cards drawn.

To approximate P(X < 7), we can use the normal distribution. We can find the mean and standard deviation of X, and then use the Z-score formula to calculate the probability.

The mean of X is given by μ = n * p, where n is the number of draws and p is the probability of drawing a red card. In this case, n = 15 and p = 3/5. So, μ = 15 * (3/5) = 9.

The standard deviation of X is given by σ = sqrt(n * p * (1 - p)). In this case, σ = sqrt(15 * (3/5) * (2/5)) = 2.1909 (rounded to 4 decimal places).

Now, we can calculate the Z-score using the formula Z = (X - μ) / σ. For X = 7, the Z-score is (7 - 9) / 2.1909 = -0.9129 (rounded to 4 decimal places).

Finally, we can use a standard normal distribution table or a calculator to find the probability that the Z-score is less than -0.9129. The approximate probability that X is less than 7 is P(X < 7) = 0.1807 (rounded to 4 decimal places), or about 18.07%.