For example, there is a new video game to market, lets call it Game 1, and there is a 0.60 probability that its competitor will produce a new game similar to it. (This 0.60 may not need to be known for the solution.) There is a second video game to market that won't have competition, so lets call it Game 2.
Below are the two decision alternatives with states of nature, and probabilities for each state of nature WITH and WITHOUT competition - in thousands of dollars):
I recommend marketing Game 1 over Game 2 because:
460*.6 = 276 (probability that a competitor will produce a new game similar) 1120*04 = 448 (without competition) = 724,000 vs 640,000 Game 2
Recommendation is to market Game 1 (which is 84,000 profit difference)
Now I am ask to use sensitivity analysis to determine what the probability of competition for Game 1 would have to be for me to change my recommended decision alternative.
I do not know how to set this up to come up with 0.7273.
So, if I'm following your problem, we can set these equations up.
Game 1:
Sum (over possibilities) Expected value $\cdot$ probability = Expected Profit, which becomes $460 \cdot 0.6 +1120 \cdot 0.4 = 724$, where the dollar values are in thousands of dollars (apparently). Now, for game 2, we have
Game 2 :
$640 \cdot 1 = 640$, a stable expected profit no matter what the other guys are doing.
Thus, it's natural to say "Well, we just guessed the other guys have a $60\%$ chance of marketing a competing game. What can happen if we admit we don't know how likely they are to do this?" Here, we just say there's some possibility that the other guys market a competing game, and we let $p$ denote this unkown probability. The same analysis says that our expected profit for Game 1 (when we admit that we don't know the value of $p$) is
$$460 \cdot p + 1120 \cdot (1-p) = 460 p + 1120 - 1120 p$$ or $$1120 - 660p$$ as our expected profit as a function of $p$. Now, we want to know how $p$-values influence our decision to choose Game 1 or Game 2. We'll set up the equation of net profit of choosing game 1 over game 2:
$$1120-660p - 640 = 480 - 660p = 0,$$ where we've set this equal to zero because we'd like to know what $p$-value gives us no reason (i.e., net profit) to choose Game 1 over Game 2. Just solve this equation for $p$ to see if this agrees with the answer you're told is correct!