Question 1190374: A parent sets up a recording device linked to a computer to figure out what keys are hit over a set amount of time. Over 10 days, a toddler is strapped into a high-chair and placed in front of the piano -- its full-sized, 88 keys, standard configuration. The parent records keys to hit for over 100 trials over the course of 10 days (1,000 keys hit total). The child never bangs on the piano, they only use a single finger. What would the parent expect the distribution of key strikes over the 10 day period to look like and what key would likely be the mode?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! The parent should expect the distribution of key strikes to be approximately **uniform**.
Here's the reasoning:
* **Randomness:** The toddler is using a single finger and presumably hitting keys somewhat randomly (or at least without a specific musical intent). While there might be some slight biases (e.g., perhaps they like hitting higher keys more often), over 1000 key strikes, these biases should largely even out.
* **Large Sample Size:** With 1000 key strikes, the law of large numbers suggests that the observed distribution will get closer and closer to the true underlying distribution. Since the key presses are assumed to be random, the true distribution should be close to uniform.
* **Equal Probability (in theory):** If the child were truly hitting keys completely at random, each of the 88 keys would have an equal probability of being struck.
**Mode:**
Since the distribution is expected to be approximately uniform, *any* key is equally likely to be the mode. In a perfectly uniform distribution, there is no mode (or all values are modes). In the real-world scenario with some slight biases, the mode would be the key that was hit *slightly* more often than the others, purely due to chance. However, it's important to note that the difference in frequency between the most-hit key and the other keys would likely be small, and the mode would not be particularly meaningful. It would be more representative of random variation than a true preference for that key.
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