shooting game

[jasen]

One day on a shooting game, you are given two choices.
1. You only have 1 chance shooting a target with open eyes.
2. You are given 3 chances shooting the target with closed eyes.
You realize that your shooting ability is only 1/3 with closed eyes than open eyes.

What choice do you take to maximize your probability to win?

 

Edited September 4, 2015 by jasen

[k-man]

It depends on how good you are with your eyes open.

Let P be the probability of scoring with your eyes open, then P/5 is the probability of scoring with your eyes closed.

1-P = probability of missing one shot with eyes open. (1-P/5)⁶ = probability of missing all 6 shots with eyes closed.

These probabilities are equal when P is approximately 0.367403, so if you're scoring better than 36.7% with your eyes open, go for a single shot. If you're worse than that then 6 shots with your eyes closed is a better choice.

P.S. Calculation results are courtesy of WolframAlpha http://www.wolframalpha.com/input/?i=1-p%3D(1-p%2F5)^6%2C+0<p<1. 

 

[bubbled]

I'm going with:

 

1 shot

If p = probability of success with eyes open -- in the range 0 < p < 9;  (1 - p) < (1 - (p/3))^3. p larger than 1 doesn't make sense.

[jasen]

I'm going with:

 

Hidden Content

Right !

Now I modifiy the question 

One day on a shooting game, you are given two options
1. You only have 1 chance shooting a target with open eyes.
2. You are given 6 chances shooting the target with closed eyes.
You realize that your shooting ability is only 1/5 with closed eyes than open eyes.

What option do you take to maximize your chance to win?

 

Edited September 5, 2015 by jasen

[jasen]

No more try to answer?

[jasen]

Hidden Content

 

K-man is right

I was thought shooting skill when open eyes is not important, but I was wrong.

I have created VBA code to test this, and this agree with K-man answer

Sub AccuracyTest()
Dim win As Single
Dim cnt As Double
Dim Shot As Integer
Const openEyes = 37 'percentage if open eyes
Const divFactor = 5  'divison factor if closed eyes
Const NumOfGames = 1000000
Const NumOfTries = 6

    'test open eyes
    win = 0
    For cnt = 1 To NumOfGames
        Shot = Int(100 * Rnd)
        If Shot < openEyes Then win = win + 1
    Next cnt
    Selection.TypeText Text:=vbCrLf + str(win / NumOfGames)
    
    'test closed eyes
    win = 0
    For cnt = 1 To NumOfGames
        For j = 1 To NumOfTries
            Shot = Int(100 * divFactor * Rnd)
            If Shot < openEyes Then
                win = win + 1
                GoTo finish
            End If
        Next j
finish:
    Next cnt
    Selection.TypeText Text:=str(win / NumOfGames)
End Sub