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Probability- Decision making

There is a 0.25 probability that the temperature will drop below freezing, and Best Orange Company has to decide on whether or not to fire up its smudge pots to protect its $9500 crop investment. It will cost the company $2000 to employ the smudge pots. If there is a freeze the crop would realize a revenue of $23,000 at market, as opposed to a revenue of $17,000 if there is no freeze (a matter of supply and demand). The crop will not survive a freeze without the smudge pots. Should the pots be used?
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3 Answers

This is a great question. To determine whether the company should fire up the pots, you need to compare the expected profit (revenue - cost) of the 2 scenarios. 
 
Scenario 1 is fire up the pots. If we fire up the pots, 
 
our expected revenue is .2(23,000)+ .8(17,000)  = 
4,600+13,600 = 18,200. 
But there is the cost of firing up the pots along w the. Original investment cost so the profit would be 18,200-(2000+9,500) =6,700
 
scenario 2
 
the profit if we don't fire up the pots is 
we only get a profit if it doesn't freeze. So the expected revenue is
.2(0)+.8(17,000)=13,600
the cost is just the cost of investment 9,500
so expected profit = 13,600-9,500=4,100
 
since the expected profit of firing up the pots is greater than not, they should fire up the pots!
The quick answer is that the additional revenue of $23,000 with a probability of 0.25 is on average 0.25*$23,000=$5,750, which is bigger than the $2,000 it costs the company to employ the smudge pots, so it is worth using the pots.
 
The long way is to find the expected revenues with and without using the smudge pots.
Without the pots, it is
E = 0.25*0 + 0.75*17,000 = $12,750
With the pots, it is
E = 0.25*23,000 + 0.75*17,000 - 2000 = $16,500
(These are the revenues, for the net profit subtract $9,500 from either number.)
 
so it is higher with the pots used.
 
Since the probability of a freeze is .25 that means if you had 4 situations just like this you would be protecting unnecessarily 3 times and protecting necessarily once. Let's compare two farmers, one that never protects with a probability of .25 and one that does. Farmer 1 Situation 1 No freeze...Makes a profit of $17,000-$9,500=$7,500 Situation 2 No freeze...Makes a profit of $17,000-$9,500=$7,500 Situation 3 No freeze...Makes a profit of $17,000-$9,500=$7,500 Situation 4 freeze...Loses $9,500 Total profit=3($7,500)-$9,500=$13,000 Farmer 2 Situation 1 No freeze...Makes a profit of $17,000-$9,500-$2,000=$5,500 Situation 2 No freeze...Makes a profit of $17,000-$9,500-$2,000=$5,500 Situation 3 No freeze...Makes a profit of $17,000-$9,500-$2,000=$5,500 Situation 4 freeze...Makes a profit of $23,000-$9,500-$2,000=$11,500 Total profit=3($5,500)+$11,500=$28,000 Protecting is a much better decision with this probability. Even if the probability was .1 it would still pay to protect. The break even point is between .0833 and .0909.

Comments

The break even point is at 2,000/23,000≈0.087.
Thanks Andre...I was originally thinking the farmer would make more if it only happened 1 out of 11 but it would only be more if it was 1 out of 12 or more.  Thanks for the easy way to determine the cost loss ratio!