A company looking for venture capitalist funding is deciding on the design of its operating system for its new phone. The first option is to simply buy the OS from another company. This would result in sales of either 10,000 units if the market is not crowded with similar phones or sales of only 3,000 units if the market is crowded. If the company decides to design its own OS the phone would have sales of 70,000 units if the OS was popular but sales of only 2,000 if the OS was a failure. Suppose that to recoup the cost of designing their own OS the company would need to sell twice as many phones as when they simply buy the OS for the profit from the scenarios to be equal. Which option should the company choose if the probability that the market is/ is not crowded is 50% and the probability that the OS is popular is 75%?

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To determine which option the company should choose, we need to compare the expected profits of each option.

Let's calculate the expected profit of buying the OS from another company:

- If the market is not crowded (50% probability), the company would sell 10,000 units with a profit of 10,000 units.
- If the market is crowded (50% probability), the company would sell 3,000 units with a profit of 3,000 units.

The expected profit of buying the OS is therefore:
(0.5 * 10,000) + (0.5 * 3,000) = 5,000 + 1,500 = 6,500 units.

Now, let's calculate the expected profit of designing their own OS:

- If the OS is popular (75% probability), the company would sell 70,000 units with a profit of 70,000 units.
- If the OS is a failure (25% probability), the company would sell 2,000 units with a profit of 2,000 units.

The expected profit of designing their own OS is therefore:
(0.75 * 70,000) + (0.25 * 2,000) = 52,500 + 500 = 53,000 units.

To make the profit from both scenarios equal, the expected profit from designing their own OS needs to be twice the expected profit from buying the OS:

53,000 units = 2 * 6,500 units.

Now we can solve this equation:

2 * 6,500 units = 53,000 units
13,000 units = 53,000 units
13,000/53,000 = 0.2453

Thus, the company should choose to design its own OS if the probability that the market is not crowded is greater than approximately 0.2453, or if the probability that the market is crowded is less than approximately 0.7547.

In this case, since the probability that the market is crowded is 50%, the company should choose to design its own OS because the probability is less than 0.7547.