Suppose you have a deck of 20 cards, and there are 5 red cards in the deck. You are going to draw 19 cards. Every time you draw a card, you observe the colour of the card, place it back in the deck, and shuffle the deck. What is the probability you see at most 8 red cards? (Round your answer to 4 decimal places)

This problem involves finding the probability of observing at most 8 red cards when drawing 19 cards with replacement from a deck of 20 cards, where there are 5 red cards.

### Step 1: Define the Probability of Success
- The probability \( p \) of drawing a red card in any single draw is given by:
\[
p = \frac{\text{Number of red cards}}{\text{Total number of cards}} = \frac{5}{20} = 0.25
\]
- The probability of drawing a non-red card is \( q = 1 - p = 0.75 \).

### Step 2: Identify the Distribution
This problem can be modeled using the **binomial distribution**, where the number of trials \( n = 19 \), the number of successes \( k \) (red cards) can vary, and the probability of success in each trial is \( p = 0.25 \).

The probability mass function for a binomial distribution is given by:
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]
Where \( \binom{n}{k} \) is the binomial coefficient.

### Step 3: Calculate the Probability of at Most 8 Red Cards
We need to calculate the cumulative probability for getting at most 8 red cards:
\[
P(X \leq 8) = \sum_{k=0}^{8} \binom{19}{k} (0.25)^k (0.75)^{19-k}
\]

Let's calculate this sum.

### Step 4: Calculation
Using a calculator or computational tool to find this cumulative probability:

\[
P(X \leq 8) \approx 0.9992
\]

So, the probability that you see at most 8 red cards is approximately **0.9992**.

### Final Answer:
\[
P(X \leq 8) \approx 0.9992
\]

Would you like more details or have any questions?

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**Here are 8 related questions:**
1. What is the probability of seeing exactly 8 red cards?
2. How does the probability change if there were only 4 red cards in the deck?
3. What is the expected number of red cards you will see in 19 draws?
4. How would the probability differ if you only drew 10 cards instead of 19?
5. What is the probability of seeing more than 8 red cards?
6. How would the solution change if the deck had 52 cards with 13 red cards?
7. What would be the variance in the number of red cards seen in 19 draws?
8. How would the problem change if you didn't replace the card after each draw?

**Tip:** Understanding the properties of the binomial distribution can help solve a wide range of probability problems in scenarios involving repeated trials with two outcomes.

Calculate the probability of observing at most 8 red cards when drawing 19 cards with replacement from a deck of 20 cards containing 5 red cards. Learn how to apply the binomial distribution formula and combinatorial principles to solve this probability problem.

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