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The image you uploaded includes a statistical probability problem involving a fair die and a random variable \( X \), which represents the number of times a 6 shows up in 7 tosses. The task involves calculating probabilities based on a binomial distribution, analyzing the skewness of the distribution, and interpreting unusual outcomes.

### Summary of parts from the problem:

**(a)** Complete the probability distribution of \( X \).
- The probability values for \( P(X = x) \) are provided, corresponding to different numbers of sixes observed in 7 rolls:
  - \( P(X = 0) = 0.2791 \)
  - \( P(X = 1) = 0.3907 \)
  - \( P(X = 2) = 0.2344 \)
  - \( P(X = 3) = 0.0781 \)
  - \( P(X = 4) = 0.0156 \)
  - \( P(X = 5) = 0.0019 \)
  - \( P(X = 6) = 0.0001 \)
  - \( P(X = 7) = 0.00000357 \)

**(b)** Analyzing the distribution:
- The distribution is described as *skewed to the right*.
- The expected value of \( X \) (mean) is 1.17 sixes, and the standard deviation is 0.99 sixes.

**(c)** Probability of getting at least 6 sixes:
- You observed 6 sixes, and the probability of this event happening again is given as \( 0.0001 \), which is extremely low.

**(d)** Interpretation of the results:
- The correct interpretation (selected as "A") is that the distribution of the random variable \( X \) is skewed to the right. This matches the fact that most of the probability mass is concentrated on fewer sixes, and getting 6 or more sixes is very rare.

Would you like to explore further details, or do you have any questions on these concepts?

### Related Questions:
1. How is the binomial distribution used to calculate the probability in part (a)?
2. What does it mean for a distribution to be skewed to the right in terms of probability?
3. Why is the probability of getting at least 6 sixes so small in part (c)?
4. How do we calculate the expected value and standard deviation of a binomial distribution like the one in this problem?
5. What would change in the probability distribution if you tossed the die more times, say 10 or 15 times?

#### Tip:
In a binomial distribution, the skewness depends on the probability of success \( p \). When \( p < 0.5 \), as in this case where \( p = \frac{1}{6} \), the distribution tends to be right-skewed.

Your statistics professor hands you a fair die, with each side having the same chance of being the uppermost face. You are asked to toss the die 7 times and count the number of times the die shows a topside of six. This count is represented by the random variable X. The task involves completing the probability distribution, analyzing its skewness, and interpreting the probability of getting a certain number of sixes.

Explore a probability problem involving the binomial distribution of rolling a fair die seven times and analyzing the distribution of the number of sixes. This problem involves calculating probabilities, understanding skewness, and interpreting rare events. Suitable for undergraduate students studying probability.

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