To solve these probability questions, we'll use the properties of the normal distribution. Given that stock returns are approximately normally distributed, we can calculate the probabilities by finding the corresponding Z-scores and then looking up these values in a standard normal distribution table or using a normal cumulative distribution function.

Given:

• Mean (μ) for S&P 500 returns = 10.42%
• Standard deviation (σ) for S&P 500 returns = 20%

1. Probability that a stock in the S&P 500 gained value in 2017
   A stock gains value if its return is greater than 0%. We need to find $P(X > 0)$.

   • Z-score for 0%:
     $Z = \frac{X - \mu}{\sigma} = \frac{0 - 10.42}{20} = \frac{-10.42}{20} = -0.521$

   Looking up the Z-score of -0.521, we find that the cumulative probability up to this Z-score is approximately 0.3015. Therefore, the probability that a stock in the S&P 500 gained value (is above 0) is: $P(X > 0) = 1 - 0.3015 = 0.6985$ So, about 69.85% of stocks in the S&P 500 gained value in 2017.

2. Probability that a stock in the S&P 500 gained 10% or more in 2017
   We want $P(X > 10)$.

   • Z-score for 10%: $Z = \frac{10 - 10.42}{20} = \frac{-0.42}{20} = -0.021$

   The cumulative probability for a Z-score of -0.021 is approximately 0.4916. Thus, the probability that a stock gained 10% or more is: $P(X > 10) = 1 - 0.4916 = 0.5084$ So, about 50.84% of stocks in the S&P 500 gained 10% or more.

3. Probability that a stock in the S&P 500 lost 20% or more in 2017
   We want $P(X < -20)$.

   • Z-score for -20%: $Z = \frac{-20 - 10.42}{20} = \frac{-30.42}{20} = -1.521$

   The cumulative probability for a Z-score of -1.521 is approximately 0.0643. Therefore: $P(X < -20) = 0.0643$ So, about 6.43% of stocks in the S&P 500 lost 20% or more in 2017.

4. Probability that a stock in the S&P 500 lost 30% or more in 2017
   We want $P(X < -30)$.

   • Z-score for -30%: $Z = \frac{-30 - 10.42}{20} = \frac{-40.42}{20} = -2.021$

   The cumulative probability for a Z-score of -2.021 is approximately 0.0217. Thus: $P(X < -30) = 0.0217$ So, about 2.17% of stocks in the S&P 500 lost 30% or more in 2017.

Summary of Probabilities:

• Probability that a stock gained value: 69.85%
• Probability that a stock gained 10% or more: 50.84%
• Probability that a stock lost 20% or more: 6.43%
• Probability that a stock lost 30% or more: 2.17%

Would you like further details on any of these steps?

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Related Questions:

1. How would the probabilities change if the standard deviation was higher or lower?
2. What is the probability that a stock in the NASDAQ gained value in 2017?
3. How can we use the normal distribution to predict stock returns in other years?
4. What is the probability of a stock gaining between 10% and 20% in the S&P 500?
5. What does a Z-score represent in the context of financial returns?

Tip: Remember, Z-scores standardize values by the mean and standard deviation, allowing you to compare probabilities across different normal distributions.