Question

You are a farmer whose farmland is cool, moist, and shady. (Assume that these growing conditions are equivalent to 80 drops/hr of water, 10% light, and a temperature of 13°C in the Gizmo.) You are thinking about planting seeds like Seed A in the Gizmo. You will turn a profit if at least 50% of your seeds become sprouts. Are you most likely to make money or lose money on this crop?

Answer 1

Define
n = number of seeds planted.
k = number of seeds that successfully become plants.

The probability of making a profit is
p =50% = 0.5
Therefore the probability of not makng a profit is
q = 1 - p = 0.5

According to the binomial distribution, the probability of k successes out of n trials is

`P(k \, of \, n) = _(n)C_(k)\,p^(k) q^(n-k) \\ where \\ _(n)C_(k) = (n!)/(k! (n-k)!)`

Create a table as shown below by using the graphing calculator.

n k P(k of n)
---- ----- ------------
4 2 0.375
8 4 0.273
10 5 0.246
12 6 0.226
16 8 0.196
20 10 0.176
50 25 0.0.112
100 50 0.08

A graph of this table shows that the probability of success i below 0.5 (50%) for all practical values of n (the number of seeds planted).

Answer: The farmer will lose money.