In 42 of the 64 years the S&P 500 finished higher after the first five days of trading. In 37 out of the 42 years the S&P 500 finished higher for the year. If a year is selected at random, what is the probability that the S&P 500 finished lower for the year?

To solve this problem, we need to find the probability that the S&P 500 finished lower for the year. Here are the important facts from the problem:

- Out of 64 years, in 42 years, the S&P 500 finished higher after the first five days of trading.
- Out of these 42 years, the S&P 500 finished higher for the year in 37 years.
- This implies that in \(64 - 42 = 22\) years, the S&P 500 did not finish higher after the first five days of trading.

Let’s break it down step by step:

1. **Step 1: Total years when the S&P 500 finished lower for the year.**
   - Out of the 42 years where the S&P 500 finished higher after the first five days, 37 of those years finished higher for the year. Therefore, the remaining \(42 - 37 = 5\) years finished lower.
   - In the other 22 years, there is no information about what happened after the first five days, but the total number of years we are considering is still 64. Since the total number of years is fixed, all the remaining 22 years must have also finished lower.
   - Therefore, the total number of years when the S&P 500 finished lower for the year is:
     \[
     5 + 22 = 27 \text{ years}.
     \]

2. **Step 2: Probability of selecting a year where the S&P 500 finished lower for the year.**
   - The total number of years is 64.
   - The number of years when the S&P 500 finished lower is 27.
   - The probability is given by:
     \[
     P(\text{lower for the year}) = \frac{27}{64}.
     \]

Thus, the probability that the S&P 500 finished lower for the year is:

\[
P(\text{lower for the year}) = \frac{27}{64} \approx 0.4219 \text{ or } 42.19\%.
\]

Would you like more details or have any questions?

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Here are 5 related questions to expand your understanding:

1. What is the probability that the S&P 500 finished higher for the year?
2. If the S&P 500 finished higher after the first five days, what is the probability it finished higher for the year?
3. What is the complement of the probability that the S&P 500 finished lower for the year?
4. How would the probability change if the total number of years was reduced to 50?
5. What is the probability that the S&P 500 did not finish higher after the first five days?

**Tip:** When calculating probabilities, always ensure that the total number of outcomes and the number of favorable outcomes are clearly defined.

Determine the probability that the S&P 500 finished lower for the year using key probability concepts. In this problem, 27 out of 64 years saw the S&P 500 finish lower, leading to a probability of approximately 42.19%. Suitable for students in Grades 10-12.

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