Create a X and Y table using the expression Y= 4X +3
2 answers:
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The maximum earning is v(r,b)
Let v(r,b) be the expected value of the game for the player, assuming optimal play, if the remaining deck has r red cards and b black cards.
Then v(r,b) satisfies the recursion
and
The stopping rule is simple: Stop when v(r,b)=0.
To explain the recursion . . .
If r,b>0, and the player elects to play a card, then:
- The revealed card is red with probability
, and in that case, the player gets a score of +1, and the new value is V(r-1,b)
- The revealed card is black with probability
, and in that case, the player gets a score of −1, and the new value is V(r,b-1)
Thus, if r,b>0, electing to play a card yields the value f(r,b).
But the player always has the option to quit, hence, if r,b>0, we get v(r,b)=max(0,f(r,b)).
Implementing the recursion in Maple, the value of the game is
v(26,26)=41984711742427/15997372030584
v(26,26) ≈2.624475549
and the optimal stopping strategy is as follows . . .
- If 24≤b≤26, play while r≥b−5.
- If 17≤b≤23, play while r≥b−4.
- If 11≤b≤16, play while r≥b−3.
- If 6≤b≤10, play while r≥b−2.
- If 3≤b≤5, play while r≥b−1.
- If 1≤b≤2, play while r≥b.
- If b=0, play while r>0.
So, The maximum earning is v(r,b)
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ANSWER
An expression for P(t) is
EXPLANATION
This is a separable differential equation. Treat M and k as constants. Then we can divide both sides by M - P to get the P term with the differential dP and multiply both sides by dt to separate dt from the P terms
Integrate both sides of the equation.
Note that for the left-hand side, u-substitution gives us
hence why
Now we use the definition of the logarithm to convert into exponential form.
The definition is
so applying it here, we get
Exponent properties can be used to address the constant C. We use
here:
If we assume that P(0) = 0, then set t = 0 and P = 0
Substituting into our original equation, we get our final answer of
ANSWER
An expression for P(t) is
EXPLANATION
This is a separable differential equation. Treat M and k as constants. Then we can divide both sides by M - P to get the P term with the differential dP and multiply both sides by dt to separate dt from the P terms
Integrate both sides of the equation.
Note that for the left-hand side, u-substitution gives us
hence why
Now we use the definition of the logarithm to convert into exponential form.
The definition is
so applying it here, we get
Exponent properties can be used to address the constant C. We use
If we assume that P(0) = 0, then set t = 0 and P = 0
Substituting into our original equation, we get our final answer of