1) 4.55
2) Short hit
Step-by-step explanation:
1)
The table containing the score and the relative probability of each score is:
Score 3 4 5 6 7
Probability 0.15 0.40 0.25 0.15 0.05
Here we call
X = Miguel's score on the Water Hole
The expected value of a certain variable X is given by:
![E(X)=\sum x_i p_i](https://tex.z-dn.net/?f=E%28X%29%3D%5Csum%20x_i%20p_i)
where
are all the possible values that the variable X can take
is the probability that ![X=x_i](https://tex.z-dn.net/?f=X%3Dx_i)
Therefore in this problem, the expected value of MIguel's score is given by:
![E(X)=3\cdot 0.15 + 4\cdot 0.40 + 5\cdot 0.25 + 6\cdot 0.15 + 7\cdot 0.05=4.55](https://tex.z-dn.net/?f=E%28X%29%3D3%5Ccdot%200.15%20%2B%204%5Ccdot%200.40%20%2B%205%5Ccdot%200.25%20%2B%206%5Ccdot%200.15%20%2B%207%5Ccdot%200.05%3D4.55)
2)
In this problem, we call:
X = Miguel's score on the Water Hole
Here we have that:
- If the long hit is successfull, the expected value of X is
![E(X)=4.2](https://tex.z-dn.net/?f=E%28X%29%3D4.2)
- Instead, if the long hit fails, the expected value of X is
![E(X)=5.4](https://tex.z-dn.net/?f=E%28X%29%3D5.4)
Here we also know that the probability of a successfull long hit is
![p(L)=0.4](https://tex.z-dn.net/?f=p%28L%29%3D0.4)
Which means that the probabilty of an unsuccessfull long hit is
![p(L^c)=1-p(L)=1-0.4=0.6](https://tex.z-dn.net/?f=p%28L%5Ec%29%3D1-p%28L%29%3D1-0.4%3D0.6)
Therefore, the expected value of X if Miguel chooses the long hit approach is:
![E(X)=p(L)\cdot 4.2 + p(L^C)\cdot 5.4 = 0.4\cdot 4.2 + 0.6\cdot 5.4 =4.92](https://tex.z-dn.net/?f=E%28X%29%3Dp%28L%29%5Ccdot%204.2%20%2B%20p%28L%5EC%29%5Ccdot%205.4%20%3D%200.4%5Ccdot%204.2%20%2B%200.6%5Ccdot%205.4%20%3D4.92)
In part 1) of the problem, we saw that the expected value for the short hit was instead
![E(X)=4.55](https://tex.z-dn.net/?f=E%28X%29%3D4.55)
Since the expected value for X is lower (=better) for the short hit approach, we can say that the short hit approach is better.